A332231 a(n) = 1/n! * ((n+1)*n)!/Gamma(1 + (n+1)*n/2) * Gamma(1 + (n-1)*n/2)/((n-1)*n)!.

1, 2, 30, 924, 41990, 2521260, 188296108, 16825310040, 1750702260294, 207921866100300, 27755558583300548, 4114068719809705800, 670456479908731386780, 119149476568133242798840, 22932161636278362035091480

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..310

Formula

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

From Vaclav Kotesovec, Feb 08 2020: (Start)

a(n) ~ 2^(n - 1/2) * exp(n) * n^(n - 1/2) / sqrt(Pi).

a(n) = binomial(n*(n+1), 2*n) * binomial(2*n, n) / binomial(n*(n+1)/2, n). (End)

Mathematica

Table[Sum[Binomial[(n + 1)*n, k]*Binomial[n^2 - k - 1, n - k], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Feb 08 2020 *)
Table[Binomial[n*(n+1), 2*n] * Binomial[2*n, n] / Binomial[n*(n+1)/2, n], {n, 0, 15}] (* Vaclav Kotesovec, Feb 08 2020 *)

Prog

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

Crossrefs

Main diagonal of A330843.

Sequence in context: A229781 A114938 A082653 * A274389 A186292 A273661

Adjacent sequences: A332228 A332229 A332230 * A332232 A332233 A332234

Keyword

nonn

Author

Seiichi Manyama, Feb 08 2020

Status

approved