# Fungrim entry: 54f420

$\sin^{2 n}\!\left(z\right) = \frac{1}{{4}^{n}} {2 n \choose n} + \frac{2}{{4}^{n}} \sum_{k=0}^{n - 1} {\left(-1\right)}^{n + k} {2 n \choose k} \cos\!\left(2 \left(n - k\right) z\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
TeX:
\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}
Definitions:
Fungrim symbol Notation Short description
Pow${a}^{b}$ Power
Sin$\sin(z)$ Sine
Binomial${n \choose k}$ Binomial coefficient
Sum$\sum_{n} f(n)$ Sum
Cos$\cos(z)$ Cosine
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("54f420"),
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)))))

## Topics using this entry

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