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Fungrim entry: 788fa4

(zk)=i=1kz+1ii=i=0k1zii+1{z \choose k} = \prod_{i=1}^{k} \frac{z + 1 - i}{i} = \prod_{i=0}^{k - 1} \frac{z - i}{i + 1}
Assumptions:zC  and  kZ0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0}
z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
Binomial(nk){n \choose k} Binomial coefficient
Productnf(n)\prod_{n} f(n) Product
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Equal(Binomial(z, k), Product(Div(Sub(Add(z, 1), i), i), For(i, 1, k)), Product(Div(Sub(z, i), Add(i, 1)), For(i, 0, Sub(k, 1)))),
    Variables(z, k),
    Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC