Semantic Graph — Visual Examination Report

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\binom{n}{k} = \\frac{n!}{k!(n-k)!}",
      "chartScript": {
        "script": "binomial(n, k) - factorial(n)/(factorial(k)*factorial(-k + n))",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__choose_2",
      "type": "function",
      "op": "choose",
      "subexpr": "{\\binom{n}{k}}",
      "chartScript": {
        "script": "binomial(n, k)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "n! \\frac{1}{k! \\left(- k + n\\right)!}",
      "chartScript": {
        "script": "factorial(n)/(factorial(k)*factorial(-k + n))",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__factorial_4",
      "type": "operator",
      "op": "factorial",
      "subexpr": "n!",
      "chartScript": {
        "script": "factorial(n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__power_5",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{k! \\left(- k + n\\right)!}",
      "chartScript": {
        "script": "1/(factorial(k)*factorial(-k + n))",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__multiply_6",
      "type": "operator",
      "op": "multiply",
      "subexpr": "k! \\left(- k + n\\right)!",
      "chartScript": {
        "script": "factorial(k)*factorial(-k + n)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__factorial_7",
      "type": "operator",
      "op": "factorial",
      "subexpr": "k!",
      "chartScript": {
        "script": "factorial(k)",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__factorial_8",
      "type": "operator",
      "op": "factorial",
      "subexpr": "\\left(- k + n\\right)!",
      "chartScript": {
        "script": "factorial(-k + n)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__add_9",
      "type": "operator",
      "op": "add",
      "subexpr": "n - k",
      "chartScript": {
        "script": "-k + n",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__negation_10",
      "type": "operator",
      "op": "negation",
      "subexpr": "-k",
      "chartScript": {
        "script": "-k",
        "variables": [
          "k"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "n",
      "to": "__choose_2"
    },
    {
      "from": "k",
      "to": "__choose_2"
    },
    {
      "from": "__choose_2",
      "to": "__equals_1"
    },
    {
      "from": "n",
      "to": "__factorial_4"
    },
    {
      "from": "__factorial_4",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "k",
      "to": "__factorial_7"
    },
    {
      "from": "__factorial_7",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "n",
      "to": "__add_9"
    },
    {
      "from": "k",
      "to": "__negation_10"
    },
    {
      "from": "__negation_10",
      "to": "__add_9"
    },
    {
      "from": "__add_9",
      "to": "__factorial_8"
    },
    {
      "from": "__factorial_8",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_6",
      "to": "__power_5"
    },
    {
      "from": "__power_5",
      "to": "__multiply_3"
    },
    {
      "from": "__multiply_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __choose_2{{"$\binom{\cdot}{\cdot}$"}}:::function
  n["$n$"]:::scalar
  k["$k$"]:::scalar
  __multiply_3(("$\times$")):::operator
  __factorial_4(("$(\cdot)!$")):::operator
  __power_5(("$\dfrac{1}{(\cdot)}$")):::operator
  __multiply_6(("$\times$")):::operator
  __factorial_7(("$(\cdot)!$")):::operator
  __factorial_8(("$(\cdot)!$")):::operator
  __add_9(("$+$")):::operator
  __negation_10@{ shape: "flip-tri", label: "$-$" }
  class __negation_10 operator
  n --> __choose_2
  k --> __choose_2
  __choose_2 --> __equals_1
  n --> __factorial_4
  __factorial_4 --> __multiply_3
  k --> __factorial_7
  __factorial_7 --> __multiply_6
  n --> __add_9
  k --> __negation_10
  __negation_10 --> __add_9
  __add_9 --> __factorial_8
  __factorial_8 --> __multiply_6
  __multiply_6 --> __power_5
  __power_5 -.-> __multiply_3
  __multiply_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#ef5350,stroke-width:3px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#ef5350,stroke-width:3px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#ef5350,stroke-width:3px
  linkStyle 12 stroke:#aaa,stroke-width:2px
  linkStyle 13 stroke:#42a5f5,stroke-width:1px
  linkStyle 14 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "a_n = a_{n-1} + a_{n-2}",
      "chartScript": {
        "script": "a_n - a_n - 1 - a_n - 2",
        "variables": [
          "a_n",
          "a_n - 1",
          "a_n - 2"
        ]
      }
    },
    {
      "id": "a_n",
      "type": "scalar",
      "latex": "a_{n}",
      "subexpr": "a_{n}",
      "chartScript": {
        "script": "a_n",
        "variables": [
          "a_n"
        ]
      }
    },
    {
      "id": "__add_2",
      "type": "operator",
      "op": "add",
      "subexpr": "a_{n - 1} + a_{n - 2}",
      "chartScript": {
        "script": "a_n - 1 + a_n - 2",
        "variables": [
          "a_n - 1",
          "a_n - 2"
        ]
      }
    },
    {
      "id": "a_n_1",
      "type": "scalar",
      "latex": "a_{n - 1}",
      "subexpr": "a_{n - 1}",
      "chartScript": {
        "script": "a_n - 1",
        "variables": [
          "a_n - 1"
        ]
      }
    },
    {
      "id": "a_n_2",
      "type": "scalar",
      "latex": "a_{n - 2}",
      "subexpr": "a_{n - 2}",
      "chartScript": {
        "script": "a_n - 2",
        "variables": [
          "a_n - 2"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "a_n",
      "to": "__equals_1"
    },
    {
      "from": "a_n_1",
      "to": "__add_2"
    },
    {
      "from": "a_n_2",
      "to": "__add_2"
    },
    {
      "from": "__add_2",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  a_n["$a_{n}$"]:::scalar
  __add_2(("$+$")):::operator
  a_n_1["$a_{n - 1}$"]:::scalar
  a_n_2["$a_{n - 2}$"]:::scalar
  a_n --> __equals_1
  a_n_1 --> __add_2
  a_n_2 --> __add_2
  __add_2 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\sum_{k=0}^{n} \\binom{n}{k} = 2^n",
      "chartScript": {
        "script": "-pow(2, n) + ((re(n) <= 0 && re(n) > -1 || re(n) > 0) ? (\n   pow(2, n)\n)\n: (\n   Sum(binomial(n, k), (k, 0, n))\n))",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__num_3",
      "type": "number",
      "label": "0",
      "subexpr": "0",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__equals_4",
      "type": "relation",
      "op": "equals",
      "subexpr": "k = 0",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__sum_2",
      "type": "operator",
      "op": "sum",
      "with_respect_to": "k",
      "lower_bound": "__num_3",
      "upper_bound": "n",
      "subexpr": "\\sum_{k=0}^{n} {\\binom{n}{k}}",
      "chartScript": {
        "script": "((re(n) <= 0 && re(n) > -1 || re(n) > 0) ? (\n   pow(2, n)\n)\n: (\n   Sum(binomial(n, k), (k, 0, n))\n))",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__choose_5",
      "type": "function",
      "op": "choose",
      "subexpr": "{\\binom{n}{k}}",
      "chartScript": {
        "script": "binomial(n, k)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__power_6",
      "type": "operator",
      "op": "power",
      "subexpr": "2^{n}",
      "chartScript": {
        "script": "pow(2, n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__num_7",
      "type": "number",
      "label": "2",
      "subexpr": "2",
      "chartScript": {
        "script": "2",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "k",
      "to": "__sum_2",
      "role": "wrt"
    },
    {
      "from": "k",
      "to": "__equals_4"
    },
    {
      "from": "__num_3",
      "to": "__equals_4"
    },
    {
      "from": "__equals_4",
      "to": "__sum_2",
      "role": "lb"
    },
    {
      "from": "n",
      "to": "__sum_2",
      "role": "ub"
    },
    {
      "from": "n",
      "to": "__choose_5"
    },
    {
      "from": "k",
      "to": "__choose_5"
    },
    {
      "from": "__choose_5",
      "to": "__sum_2"
    },
    {
      "from": "__sum_2",
      "to": "__equals_1"
    },
    {
      "from": "__num_7",
      "to": "__power_6"
    },
    {
      "from": "n",
      "to": "__power_6",
      "role": "exp"
    },
    {
      "from": "__power_6",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  k["$k$"]:::scalar
  __num_3["$0$"]:::number
  n["$n$"]:::scalar
  __equals_4{"="}:::relation
  __sum_2(("$\sum_{k}$")):::operator
  __choose_5{{"$\binom{\cdot}{\cdot}$"}}:::function
  __power_6(("$(\cdot)^{\cdot}$")):::operator
  __num_7["$2$"]:::number
  k -->|wrt| __sum_2
  k --> __equals_4
  __num_3 --> __equals_4
  __equals_4 -->|lb| __sum_2
  n -->|ub| __sum_2
  n --> __choose_5
  k --> __choose_5
  __choose_5 --> __sum_2
  __sum_2 --> __equals_1
  __num_7 --> __power_6
  n -->|exp| __power_6
  __power_6 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "n! = \\sqrt{2\\pi n} \\left(\\frac{n}{e}\\right)^n",
      "chartScript": {
        "script": "-sqrt(2)*sqrt(pi)*sqrt(n)*pow(n*exp(-1), n) + factorial(n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__factorial_2",
      "type": "operator",
      "op": "factorial",
      "subexpr": "n!",
      "chartScript": {
        "script": "factorial(n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\sqrt{2 n \\pi} \\left(\\frac{n}{e}\\right)^{n}",
      "chartScript": {
        "script": "sqrt(2)*sqrt(pi)*sqrt(n)*pow(n*exp(-1), n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__power_4",
      "type": "operator",
      "op": "power",
      "exponent": "1/2",
      "subexpr": "\\sqrt{2 n \\pi}",
      "chartScript": {
        "script": "sqrt(2)*sqrt(pi)*sqrt(n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__multiply_5",
      "type": "operator",
      "op": "multiply",
      "subexpr": "2 n \\pi",
      "chartScript": {
        "script": "2*pi*n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__num_6",
      "type": "number",
      "label": "2",
      "subexpr": "2",
      "chartScript": {
        "script": "2",
        "variables": []
      }
    },
    {
      "id": "__multiply_7",
      "type": "operator",
      "op": "multiply",
      "subexpr": "n \\pi",
      "chartScript": {
        "script": "pi*n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "pi",
      "type": "constant",
      "latex": "\\pi",
      "subexpr": "\\pi",
      "chartScript": {
        "script": "pi",
        "variables": []
      }
    },
    {
      "id": "__power_8",
      "type": "operator",
      "op": "power",
      "subexpr": "\\left(\\frac{n}{e}\\right)^{n}",
      "chartScript": {
        "script": "pow(n*exp(-1), n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__multiply_9",
      "type": "operator",
      "op": "multiply",
      "subexpr": "n \\frac{1}{e}",
      "chartScript": {
        "script": "n*exp(-1)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__power_10",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{e}",
      "chartScript": {
        "script": "exp(-1)",
        "variables": []
      }
    },
    {
      "id": "e",
      "type": "scalar",
      "latex": "e",
      "subexpr": "e",
      "chartScript": {
        "script": "e",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "n",
      "to": "__factorial_2"
    },
    {
      "from": "__factorial_2",
      "to": "__equals_1"
    },
    {
      "from": "__num_6",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "pi",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "n",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_7",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_5",
      "to": "__power_4"
    },
    {
      "from": "__power_4",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "n",
      "to": "__multiply_9",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "e",
      "to": "__power_10"
    },
    {
      "from": "__power_10",
      "to": "__multiply_9"
    },
    {
      "from": "__multiply_9",
      "to": "__power_8"
    },
    {
      "from": "n",
      "to": "__power_8",
      "role": "exp"
    },
    {
      "from": "__power_8",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __factorial_2(("$(\cdot)!$")):::operator
  n["$n$"]:::scalar
  __multiply_3(("$\times$")):::operator
  __power_4(("${(\cdot)}^{1/2}$")):::operator
  __multiply_5(("$\times$")):::operator
  __num_6["$2$"]:::number
  __multiply_7(("$\times$")):::operator
  pi["$\pi$"]:::constant
  __power_8(("$(\cdot)^{\cdot}$")):::operator
  __multiply_9(("$\times$")):::operator
  __power_10(("$\dfrac{1}{(\cdot)}$")):::operator
  e["$e$"]:::scalar
  n --> __factorial_2
  __factorial_2 --> __equals_1
  __num_6 --> __multiply_5
  pi --> __multiply_7
  n --> __multiply_7
  __multiply_7 --> __multiply_5
  __multiply_5 --> __power_4
  __power_4 --> __multiply_3
  n --> __multiply_9
  e --> __power_10
  __power_10 -.-> __multiply_9
  __multiply_9 --> __power_8
  n -->|exp| __power_8
  __power_8 --> __multiply_3
  __multiply_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#ef5350,stroke-width:3px
  linkStyle 3 stroke:#ef5350,stroke-width:3px
  linkStyle 4 stroke:#ef5350,stroke-width:3px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#ef5350,stroke-width:3px
  linkStyle 8 stroke:#ef5350,stroke-width:3px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#42a5f5,stroke-width:1px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#aaa,stroke-width:2px
  linkStyle 13 stroke:#ef5350,stroke-width:3px
  linkStyle 14 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__ceiling_1",
      "type": "function",
      "op": "ceiling",
      "subexpr": "\\lceil n/k \\rceil",
      "chartScript": {
        "script": "ceil(n/k)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__multiply_2",
      "type": "operator",
      "op": "multiply",
      "subexpr": "n \\frac{1}{k}",
      "chartScript": {
        "script": "n/k",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__power_3",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{k}",
      "chartScript": {
        "script": "1/k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "n",
      "to": "__multiply_2",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "k",
      "to": "__power_3"
    },
    {
      "from": "__power_3",
      "to": "__multiply_2"
    },
    {
      "from": "__multiply_2",
      "to": "__ceiling_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __ceiling_1{{"$ceiling(\cdot)$"}}:::function
  __multiply_2(("$\times$")):::operator
  n["$n$"]:::scalar
  __power_3(("$\dfrac{1}{(\cdot)}$")):::operator
  k["$k$"]:::scalar
  n --> __multiply_2
  k --> __power_3
  __power_3 -.-> __multiply_2
  __multiply_2 --> __ceiling_1
  linkStyle 0 stroke:#ef5350,stroke-width:3px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#42a5f5,stroke-width:1px
  linkStyle 3 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "G(x) = \\sum_{n=0}^{\\infty} a_n x^n",
      "chartScript": {
        "script": "-a_n*((abs(x) < 1) ? (\n   1/(1 - x)\n)\n: (\n   Sum(pow(x, n), (n, 0, Number.POSITIVE_INFINITY))\n)) + G(x)",
        "variables": [
          "a_n",
          "x"
        ]
      }
    },
    {
      "id": "__G_2",
      "type": "function",
      "op": "G",
      "subexpr": "G{\\left(x \\right)}",
      "chartScript": {
        "script": "G*x",
        "variables": [
          "G",
          "x"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__num_4",
      "type": "number",
      "label": "0",
      "subexpr": "0",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__const_5",
      "type": "constant",
      "label": "infinity",
      "latex": "\\infty",
      "subexpr": "\\infty",
      "chartScript": {
        "script": "Number.POSITIVE_INFINITY",
        "variables": []
      }
    },
    {
      "id": "__equals_6",
      "type": "relation",
      "op": "equals",
      "subexpr": "n = 0",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__sum_3",
      "type": "operator",
      "op": "sum",
      "with_respect_to": "n",
      "lower_bound": "__num_4",
      "upper_bound": "__const_5",
      "subexpr": "\\sum_{n=0}^{\\infty} a_{n} x^{n}",
      "chartScript": {
        "script": "a_n*((abs(x) < 1) ? (\n   1/(1 - x)\n)\n: (\n   Sum(pow(x, n), (n, 0, Number.POSITIVE_INFINITY))\n))",
        "variables": [
          "a_n",
          "x"
        ]
      }
    },
    {
      "id": "__multiply_7",
      "type": "operator",
      "op": "multiply",
      "subexpr": "x^{n} a_{n}",
      "chartScript": {
        "script": "a_n*pow(x, n)",
        "variables": [
          "a_n",
          "n",
          "x"
        ]
      }
    },
    {
      "id": "a_n",
      "type": "scalar",
      "latex": "a_{n}",
      "subexpr": "a_{n}",
      "chartScript": {
        "script": "a_n",
        "variables": [
          "a_n"
        ]
      }
    },
    {
      "id": "__power_8",
      "type": "operator",
      "op": "power",
      "subexpr": "x^{n}",
      "chartScript": {
        "script": "pow(x, n)",
        "variables": [
          "n",
          "x"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "x",
      "to": "__G_2"
    },
    {
      "from": "__G_2",
      "to": "__equals_1"
    },
    {
      "from": "n",
      "to": "__sum_3",
      "role": "wrt"
    },
    {
      "from": "n",
      "to": "__equals_6"
    },
    {
      "from": "__num_4",
      "to": "__equals_6"
    },
    {
      "from": "__equals_6",
      "to": "__sum_3",
      "role": "lb"
    },
    {
      "from": "__const_5",
      "to": "__sum_3",
      "role": "ub"
    },
    {
      "from": "a_n",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "x",
      "to": "__power_8"
    },
    {
      "from": "n",
      "to": "__power_8",
      "role": "exp"
    },
    {
      "from": "__power_8",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_7",
      "to": "__sum_3"
    },
    {
      "from": "__sum_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __G_2{{"$G(\cdot)$"}}:::function
  x["$x$"]:::scalar
  n["$n$"]:::scalar
  __num_4["$0$"]:::number
  __const_5["$\infty$"]:::constant
  __equals_6{"="}:::relation
  __sum_3(("$\sum_{n}$")):::operator
  __multiply_7(("$\times$")):::operator
  a_n["$a_{n}$"]:::scalar
  __power_8(("$(\cdot)^{\cdot}$")):::operator
  x --> __G_2
  __G_2 --> __equals_1
  n -->|wrt| __sum_3
  n --> __equals_6
  __num_4 --> __equals_6
  __equals_6 -->|lb| __sum_3
  __const_5 -->|ub| __sum_3
  a_n --> __multiply_7
  x --> __power_8
  n -->|exp| __power_8
  __power_8 --> __multiply_7
  __multiply_7 --> __sum_3
  __sum_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#ef5350,stroke-width:3px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#ef5350,stroke-width:3px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "C_n = \\frac{1}{n+1} \\binom{2n}{n}",
      "chartScript": {
        "script": "C_n - binomial(2*n, n)/(n + 1)",
        "variables": [
          "C_n",
          "n"
        ]
      }
    },
    {
      "id": "C_n",
      "type": "scalar",
      "latex": "C_{n}",
      "subexpr": "C_{n}",
      "chartScript": {
        "script": "C_n",
        "variables": [
          "C_n"
        ]
      }
    },
    {
      "id": "__multiply_2",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\frac{1}{n + 1} {\\binom{2 n}{n}}",
      "chartScript": {
        "script": "binomial(2*n, n)/(n + 1)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__power_3",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{n + 1}",
      "chartScript": {
        "script": "1/(n + 1)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__add_4",
      "type": "operator",
      "op": "add",
      "subexpr": "n + 1",
      "chartScript": {
        "script": "n + 1",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__num_5",
      "type": "number",
      "label": "1",
      "subexpr": "1",
      "chartScript": {
        "script": "1",
        "variables": []
      }
    },
    {
      "id": "__choose_6",
      "type": "function",
      "op": "choose",
      "subexpr": "{\\binom{2 n}{n}}",
      "chartScript": {
        "script": "binomial(2*n, n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__multiply_7",
      "type": "operator",
      "op": "multiply",
      "subexpr": "n 2",
      "chartScript": {
        "script": "2*n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__num_8",
      "type": "number",
      "label": "2",
      "subexpr": "2",
      "chartScript": {
        "script": "2",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "C_n",
      "to": "__equals_1"
    },
    {
      "from": "n",
      "to": "__add_4"
    },
    {
      "from": "__num_5",
      "to": "__add_4"
    },
    {
      "from": "__add_4",
      "to": "__power_3"
    },
    {
      "from": "__power_3",
      "to": "__multiply_2"
    },
    {
      "from": "__num_8",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "n",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_7",
      "to": "__choose_6"
    },
    {
      "from": "n",
      "to": "__choose_6"
    },
    {
      "from": "__choose_6",
      "to": "__multiply_2",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_2",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  C_n["$C_{n}$"]:::scalar
  __multiply_2(("$\times$")):::operator
  __power_3(("$\dfrac{1}{(\cdot)}$")):::operator
  __add_4(("$+$")):::operator
  n["$n$"]:::scalar
  __num_5["$1$"]:::number
  __choose_6{{"$\binom{\cdot}{\cdot}$"}}:::function
  __multiply_7(("$\times$")):::operator
  __num_8["$2$"]:::number
  C_n --> __equals_1
  n --> __add_4
  __num_5 --> __add_4
  __add_4 --> __power_3
  __power_3 -.-> __multiply_2
  __num_8 --> __multiply_7
  n --> __multiply_7
  __multiply_7 --> __choose_6
  n --> __choose_6
  __choose_6 --> __multiply_2
  __multiply_2 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#42a5f5,stroke-width:1px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#ef5350,stroke-width:3px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#ef5350,stroke-width:3px
  linkStyle 10 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "|A \\cup B|  =  |A| + |B| - |A \\cap B|",
      "chartScript": {
        "script": "-abs(A) - abs(B) + abs(A*B*cap) + abs(A*B*cup)",
        "variables": [
          "A",
          "B",
          "cap",
          "cup"
        ]
      }
    },
    {
      "id": "__abs_2",
      "type": "function",
      "op": "abs",
      "subexpr": "\\left|{A \\cup B}\\right|"
    },
    {
      "id": "A",
      "type": "scalar",
      "latex": "A",
      "subexpr": "A",
      "chartScript": {
        "script": "A",
        "variables": [
          "A"
        ]
      }
    },
    {
      "id": "B",
      "type": "scalar",
      "latex": "B",
      "subexpr": "B",
      "chartScript": {
        "script": "B",
        "variables": [
          "B"
        ]
      }
    },
    {
      "id": "__union_3",
      "type": "operator",
      "emoji": "∪",
      "op": "union",
      "subexpr": "A \\cup B",
      "chartScript": {
        "script": "A*B*cup",
        "variables": [
          "A",
          "B",
          "cup"
        ]
      }
    },
    {
      "id": "__add_4",
      "type": "operator",
      "op": "add",
      "subexpr": "\\left|{A}\\right| + \\left|{B}\\right| - \\left|{A \\cap B}\\right|"
    },
    {
      "id": "__add_5",
      "type": "operator",
      "op": "add",
      "subexpr": "\\left|{A}\\right| + \\left|{B}\\right|"
    },
    {
      "id": "__abs_6",
      "type": "function",
      "op": "abs",
      "subexpr": "\\left|{A}\\right|"
    },
    {
      "id": "__abs_7",
      "type": "function",
      "op": "abs",
      "subexpr": "\\left|{B}\\right|"
    },
    {
      "id": "__negation_8",
      "type": "operator",
      "op": "negation",
      "subexpr": "-\\left|{A \\cap B}\\right|"
    },
    {
      "id": "__abs_9",
      "type": "function",
      "op": "abs",
      "subexpr": "\\left|{A \\cap B}\\right|"
    },
    {
      "id": "__intersection_10",
      "type": "operator",
      "emoji": "∩",
      "op": "intersection",
      "subexpr": "A \\cap B",
      "chartScript": {
        "script": "A*B*cap",
        "variables": [
          "A",
          "B",
          "cap"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "A",
      "to": "__union_3"
    },
    {
      "from": "B",
      "to": "__union_3"
    },
    {
      "from": "__union_3",
      "to": "__abs_2"
    },
    {
      "from": "__abs_2",
      "to": "__equals_1"
    },
    {
      "from": "A",
      "to": "__abs_6"
    },
    {
      "from": "__abs_6",
      "to": "__add_5"
    },
    {
      "from": "B",
      "to": "__abs_7"
    },
    {
      "from": "__abs_7",
      "to": "__add_5"
    },
    {
      "from": "__add_5",
      "to": "__add_4"
    },
    {
      "from": "A",
      "to": "__intersection_10"
    },
    {
      "from": "B",
      "to": "__intersection_10"
    },
    {
      "from": "__intersection_10",
      "to": "__abs_9"
    },
    {
      "from": "__abs_9",
      "to": "__negation_8"
    },
    {
      "from": "__negation_8",
      "to": "__add_4"
    },
    {
      "from": "__add_4",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __abs_2{{"$|\cdot|$"}}:::function
  A["$A$"]:::scalar
  B["$B$"]:::scalar
  __union_3(("$\cup$")):::operator
  __add_4(("$+$")):::operator
  __add_5(("$+$")):::operator
  __abs_6{{"$|\cdot|$"}}:::function
  __abs_7{{"$|\cdot|$"}}:::function
  __negation_8@{ shape: "flip-tri", label: "$-$" }
  __abs_9{{"$|\cdot|$"}}:::function
  __intersection_10(("$\cap$")):::operator
  class __negation_8 operator
  A --> __union_3
  B --> __union_3
  __union_3 --> __abs_2
  __abs_2 --> __equals_1
  A --> __abs_6
  __abs_6 --> __add_5
  B --> __abs_7
  __abs_7 --> __add_5
  __add_5 --> __add_4
  A --> __intersection_10
  B --> __intersection_10
  __intersection_10 --> __abs_9
  __abs_9 --> __negation_8
  __negation_8 --> __add_4
  __add_4 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#aaa,stroke-width:2px
  linkStyle 13 stroke:#aaa,stroke-width:2px
  linkStyle 14 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\binom{n}{k} = \\binom{n-1}{k-1} + \\binom{n-1}{k}",
      "chartScript": {
        "script": "binomial(n, k) - binomial(n - 1, k) - binomial(n - 1, k - 1)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__choose_2",
      "type": "function",
      "op": "choose",
      "subexpr": "{\\binom{n}{k}}",
      "chartScript": {
        "script": "binomial(n, k)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__add_3",
      "type": "operator",
      "op": "add",
      "subexpr": "{\\binom{n - 1}{k}} + {\\binom{n - 1}{k - 1}}",
      "chartScript": {
        "script": "binomial(n - 1, k) + binomial(n - 1, k - 1)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__choose_4",
      "type": "function",
      "op": "choose",
      "subexpr": "{\\binom{n - 1}{k - 1}}",
      "chartScript": {
        "script": "binomial(n - 1, k - 1)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__add_5",
      "type": "operator",
      "op": "add",
      "subexpr": "n - 1",
      "chartScript": {
        "script": "n - 1",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__num_6",
      "type": "number",
      "label": "-1",
      "subexpr": "-1",
      "chartScript": {
        "script": "-1",
        "variables": []
      }
    },
    {
      "id": "__add_7",
      "type": "operator",
      "op": "add",
      "subexpr": "k - 1",
      "chartScript": {
        "script": "k - 1",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__num_8",
      "type": "number",
      "label": "-1",
      "subexpr": "-1",
      "chartScript": {
        "script": "-1",
        "variables": []
      }
    },
    {
      "id": "__choose_9",
      "type": "function",
      "op": "choose",
      "subexpr": "{\\binom{n - 1}{k}}",
      "chartScript": {
        "script": "binomial(n - 1, k)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "__add_10",
      "type": "operator",
      "op": "add",
      "subexpr": "n - 1",
      "chartScript": {
        "script": "n - 1",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__num_11",
      "type": "number",
      "label": "-1",
      "subexpr": "-1",
      "chartScript": {
        "script": "-1",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "n",
      "to": "__choose_2"
    },
    {
      "from": "k",
      "to": "__choose_2"
    },
    {
      "from": "__choose_2",
      "to": "__equals_1"
    },
    {
      "from": "n",
      "to": "__add_5"
    },
    {
      "from": "__num_6",
      "to": "__add_5"
    },
    {
      "from": "__add_5",
      "to": "__choose_4"
    },
    {
      "from": "k",
      "to": "__add_7"
    },
    {
      "from": "__num_8",
      "to": "__add_7"
    },
    {
      "from": "__add_7",
      "to": "__choose_4"
    },
    {
      "from": "__choose_4",
      "to": "__add_3"
    },
    {
      "from": "n",
      "to": "__add_10"
    },
    {
      "from": "__num_11",
      "to": "__add_10"
    },
    {
      "from": "__add_10",
      "to": "__choose_9"
    },
    {
      "from": "k",
      "to": "__choose_9"
    },
    {
      "from": "__choose_9",
      "to": "__add_3"
    },
    {
      "from": "__add_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __choose_2{{"$\binom{\cdot}{\cdot}$"}}:::function
  n["$n$"]:::scalar
  k["$k$"]:::scalar
  __add_3(("$+$")):::operator
  __choose_4{{"$\binom{\cdot}{\cdot}$"}}:::function
  __add_5(("$+$")):::operator
  __num_6["$-1$"]:::number
  __add_7(("$+$")):::operator
  __num_8["$-1$"]:::number
  __choose_9{{"$\binom{\cdot}{\cdot}$"}}:::function
  __add_10(("$+$")):::operator
  __num_11["$-1$"]:::number
  n --> __choose_2
  k --> __choose_2
  __choose_2 --> __equals_1
  n --> __add_5
  __num_6 --> __add_5
  __add_5 --> __choose_4
  k --> __add_7
  __num_8 --> __add_7
  __add_7 --> __choose_4
  __choose_4 --> __add_3
  n --> __add_10
  __num_11 --> __add_10
  __add_10 --> __choose_9
  k --> __choose_9
  __choose_9 --> __add_3
  __add_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#aaa,stroke-width:2px
  linkStyle 13 stroke:#aaa,stroke-width:2px
  linkStyle 14 stroke:#aaa,stroke-width:2px
  linkStyle 15 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "D_n = n! \\sum_{k=0}^{n} \\frac{(-1)^k}{k!}",
      "chartScript": {
        "script": "D_n - (-pow(-1, n + 1)*(n + 1)*exp(-1)*exp(I*pi*(-n - 1))*lowergamma(n + 1, exp_polar(I*pi))/gamma(n + 2) + exp(-1))*factorial(n)",
        "variables": [
          "D_n",
          "n"
        ]
      }
    },
    {
      "id": "D_n",
      "type": "scalar",
      "latex": "D_{n}",
      "subexpr": "D_{n}",
      "chartScript": {
        "script": "D_n",
        "variables": [
          "D_n"
        ]
      }
    },
    {
      "id": "__multiply_2",
      "type": "operator",
      "op": "multiply",
      "subexpr": "n! \\sum_{k=0}^{n} \\frac{\\left(-1\\right)^{k}}{k!}",
      "chartScript": {
        "script": "(-pow(-1, n + 1)*(n + 1)*exp(-1)*exp(I*pi*(-n - 1))*lowergamma(n + 1, exp_polar(I*pi))/gamma(n + 2) + exp(-1))*factorial(n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__factorial_3",
      "type": "operator",
      "op": "factorial",
      "subexpr": "n!",
      "chartScript": {
        "script": "factorial(n)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__num_5",
      "type": "number",
      "label": "0",
      "subexpr": "0",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__equals_6",
      "type": "relation",
      "op": "equals",
      "subexpr": "k = 0",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__sum_4",
      "type": "operator",
      "op": "sum",
      "with_respect_to": "k",
      "lower_bound": "__num_5",
      "upper_bound": "n",
      "subexpr": "\\sum_{k=0}^{n} \\frac{\\left(-1\\right)^{k}}{k!}",
      "chartScript": {
        "script": "-pow(-1, n + 1)*(n + 1)*exp(-1)*exp(I*pi*(-n - 1))*lowergamma(n + 1, exp_polar(I*pi))/gamma(n + 2) + exp(-1)",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__multiply_7",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\left(-1\\right)^{k} \\frac{1}{k!}",
      "chartScript": {
        "script": "pow(-1, k)/factorial(k)",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__power_8",
      "type": "operator",
      "op": "power",
      "subexpr": "\\left(-1\\right)^{k}",
      "chartScript": {
        "script": "pow(-1, k)",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__num_9",
      "type": "number",
      "label": "-1",
      "subexpr": "-1",
      "chartScript": {
        "script": "-1",
        "variables": []
      }
    },
    {
      "id": "__power_10",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{k!}",
      "chartScript": {
        "script": "1/factorial(k)",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__factorial_11",
      "type": "operator",
      "op": "factorial",
      "subexpr": "k!",
      "chartScript": {
        "script": "factorial(k)",
        "variables": [
          "k"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "D_n",
      "to": "__equals_1"
    },
    {
      "from": "n",
      "to": "__factorial_3"
    },
    {
      "from": "__factorial_3",
      "to": "__multiply_2",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "k",
      "to": "__sum_4",
      "role": "wrt"
    },
    {
      "from": "k",
      "to": "__equals_6"
    },
    {
      "from": "__num_5",
      "to": "__equals_6"
    },
    {
      "from": "__equals_6",
      "to": "__sum_4",
      "role": "lb"
    },
    {
      "from": "n",
      "to": "__sum_4",
      "role": "ub"
    },
    {
      "from": "__num_9",
      "to": "__power_8"
    },
    {
      "from": "k",
      "to": "__power_8",
      "role": "exp"
    },
    {
      "from": "__power_8",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "k",
      "to": "__factorial_11"
    },
    {
      "from": "__factorial_11",
      "to": "__power_10"
    },
    {
      "from": "__power_10",
      "to": "__multiply_7"
    },
    {
      "from": "__multiply_7",
      "to": "__sum_4"
    },
    {
      "from": "__sum_4",
      "to": "__multiply_2",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_2",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  D_n["$D_{n}$"]:::scalar
  __multiply_2(("$\times$")):::operator
  __factorial_3(("$(\cdot)!$")):::operator
  n["$n$"]:::scalar
  k["$k$"]:::scalar
  __num_5["$0$"]:::number
  __equals_6{"="}:::relation
  __sum_4(("$\sum_{k}$")):::operator
  __multiply_7(("$\times$")):::operator
  __power_8(("$(\cdot)^{\cdot}$")):::operator
  __num_9["$-1$"]:::number
  __power_10(("$\dfrac{1}{(\cdot)}$")):::operator
  __factorial_11(("$(\cdot)!$")):::operator
  D_n --> __equals_1
  n --> __factorial_3
  __factorial_3 --> __multiply_2
  k -->|wrt| __sum_4
  k --> __equals_6
  __num_5 --> __equals_6
  __equals_6 -->|lb| __sum_4
  n -->|ub| __sum_4
  __num_9 --> __power_8
  k -->|exp| __power_8
  __power_8 --> __multiply_7
  k --> __factorial_11
  __factorial_11 --> __power_10
  __power_10 -.-> __multiply_7
  __multiply_7 --> __sum_4
  __sum_4 --> __multiply_2
  __multiply_2 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#ef5350,stroke-width:3px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#ef5350,stroke-width:3px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#aaa,stroke-width:2px
  linkStyle 13 stroke:#42a5f5,stroke-width:1px
  linkStyle 14 stroke:#aaa,stroke-width:2px
  linkStyle 15 stroke:#ef5350,stroke-width:3px
  linkStyle 16 stroke:#aaa,stroke-width:2px