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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jul 26 2020
STATUS
approved