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Fungrim entry: 674afa

P2n ⁣(0)=(1)n4n(2nn)P_{2 n}\!\left(0\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n}
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
TeX:
P_{2 n}\!\left(0\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
Powab{a}^{b} Power
Binomial(nk){n \choose k} Binomial coefficient
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("674afa"),
    Formula(Equal(LegendrePolynomial(Mul(2, n), 0), Mul(Div(Pow(-1, n), Pow(4, n)), Binomial(Mul(2, n), n)))),
    Variables(n),
    Assumptions(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.

2019-08-17 11:32:46.829430 UTC