A331657 a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * binomial(n+k,k) * n^k.

1, 1, 13, 305, 10321, 458649, 25289461, 1666406209, 127779121345, 11178899075537, 1098961472475901, 119937806278590321, 14389588419704763409, 1882432013890951832425, 266678501426944160023653, 40673387011956179149166849, 6644919093900517186643470081

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..322

Eric Weisstein's World of Mathematics, Legendre Polynomial

Formula

a(n) = central coefficient of (1 + (2*n - 1)*x + n*(n - 1)*x^2)^n.

a(n) = [x^n] 1 / sqrt(1 - 2*(2*n - 1)*x + x^2).

a(n) = n! * [x^n] exp((2*n - 1)*x) * BesselI(0,2*sqrt(n*(n - 1))*x).

a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * (n - 1)^(n - k).

a(n) = P_n(2*n-1), where P_n is n-th Legendre polynomial.

a(n) = (-1)^n * 2F1(-n, n + 1; 1; n).

a(n) ~ 4^n * n^(n - 1/2) / (exp(1/2) * sqrt(Pi)). - Vaclav Kotesovec, Jan 26 2020

Mathematica

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[n + k, k] n^k, {k, 0, n}], {n, 1, 16}]]
Table[SeriesCoefficient[1/Sqrt[1 - 2 (2 n - 1) x + x^2], {x, 0, n}], {n, 0, 16}]
Table[LegendreP[n, 2 n - 1], {n, 0, 16}]
Table[(-1)^n Hypergeometric2F1[-n, n + 1, 1, n], {n, 0, 16}]

Prog

(PARI) a(n) = {sum(k=0, n, (-1)^(n - k) * binomial(n, k) * binomial(n+k, k) * n^k)} \\ Andrew Howroyd, Jan 23 2020
(Magma) [&+[(-1)^(n-k)*Binomial(n, k)*Binomial(n+k, k)*n^k:k in [0..n]]:n in [0..16]]; // Marius A. Burtea, Jan 23 2020

Crossrefs

Cf. A001850, A006442, A084768, A084769, A110129, A331656.

Sequence in context: A296319 A009119 A322734 * A142425 A234236 A183522

Adjacent sequences: A331654 A331655 A331656 * A331658 A331659 A331660

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 23 2020

Status

approved