A336495 - OEIS

%I #45 Aug 08 2023 07:58:02

%S 1,1,4,34,484,9946,270314,9189776,376223992,18046839982,993655820512,

%T 61803730636506,4287521490060780,328324625277864008,

%U 27514775912958768464,2505202120094546731584,246288599061132553970160,26004541628560046316399382,2935176211106696031739535696

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

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

%H Seiichi Manyama, <a href="/A336495/b336495.txt">Table of n, a(n) for n = 0..339</a>

%F 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).

%F 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

%F a(n) ~ exp(n + 3/2) * n^(n - 5/2) / sqrt(2*Pi). - _Vaclav Kotesovec_, Jul 31 2021

%F 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

%t 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 *)

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

%o (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);

%Y Main diagonal of A336573.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jul 26 2020