A336495 a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

1, 1, 4, 34, 484, 9946, 270314, 9189776, 376223992, 18046839982, 993655820512, 61803730636506, 4287521490060780, 328324625277864008, 27514775912958768464, 2505202120094546731584, 246288599061132553970160, 26004541628560046316399382, 2935176211106696031739535696

Offset

0,3

Comments

Number of Sylvester classes of n-packed words of degree n.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..339

Formula

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

a(n) = (-1)^n*binomial(1 + n^2, n)*hypergeom[-n, 1 + n^2, 2 + (n - 1)*n, 2] / (1 + n^2). - Peter Luschny, Jul 26 2020

a(n) ~ exp(n + 3/2) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Jul 31 2021

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial((n+1)*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 08 2023

Mathematica

a[n_] := ((-1)^n Binomial[1 + n^2, n] Hypergeometric2F1[-n, 1 + n^2, 2 + (n - 1) n, 2]) / (1 + n^2); Array[a, 19, 0] (* Peter Luschny, Jul 26 2020 *)

Prog

(PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(n^2+k+1, n)/(n^2+k+1));
(PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(n^2+1, k)*binomial((n+1)*n-k, n-k))/(n^2+1);

Crossrefs

Main diagonal of A336573.

Sequence in context: A198976 A156325 A248654 * A111169 A274244 A002105

Adjacent sequences: A336492 A336493 A336494 * A336496 A336497 A336498

Keyword

nonn

Author

Seiichi Manyama, Jul 26 2020

Status

approved