A331656 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^k.

1, 3, 37, 847, 28401, 1256651, 69125869, 4548342975, 348434664769, 30463322582899, 2993348092318101, 326572612514776079, 39170287549040392369, 5123157953193993402171, 725662909285939100555101, 110662236267661479984580351, 18077209893508013563092846849

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..321

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) ~ exp(1/2) * 4^n * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Jan 28 2020

Mathematica

Join[{1}, Table[Sum[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[Hypergeometric2F1[-n, n + 1, 1, -n], {n, 0, 16}]

Prog

(PARI) a(n) = {sum(k=0, n, binomial(n, k) * binomial(n+k, k) * n^k)} \\ Andrew Howroyd, Jan 23 2020

Crossrefs

Cf. A001850, A006442, A084768, A084769, A110129, A331657.

Sequence in context: A362672 A143639 A143412 * A003717 A354020 A201697

Adjacent sequences: A331653 A331654 A331655 * A331657 A331658 A331659

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 23 2020

Status

approved