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Fungrim entry: 54f420

sin2n ⁣(z)=14n(2nn)+24nk=0n1(1)n+k(2nk)cos ⁣(2(nk)z)\sin^{2 n}\!\left(z\right) = \frac{1}{{4}^{n}} {2 n \choose n} + \frac{2}{{4}^{n}} \sum_{k=0}^{n - 1} {\left(-1\right)}^{n + k} {2 n \choose k} \cos\!\left(2 \left(n - k\right) z\right)
Assumptions:zC  and  nZ0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
\sin^{2 n}\!\left(z\right) = \frac{1}{{4}^{n}} {2 n \choose n} + \frac{2}{{4}^{n}} \sum_{k=0}^{n - 1} {\left(-1\right)}^{n + k} {2 n \choose k} \cos\!\left(2 \left(n - k\right) z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
Powab{a}^{b} Power
Sinsin(z)\sin(z) Sine
Binomial(nk){n \choose k} Binomial coefficient
Sumnf(n)\sum_{n} f(n) Sum
Coscos(z)\cos(z) Cosine
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(Pow(Sin(z), Mul(2, n)), Add(Mul(Div(1, Pow(4, n)), Binomial(Mul(2, n), n)), Mul(Div(2, Pow(4, n)), Sum(Mul(Mul(Pow(-1, Add(n, k)), Binomial(Mul(2, n), k)), Cos(Mul(Mul(2, Sub(n, k)), z))), For(k, 0, Sub(n, 1))))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, 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