A383883 a(n) = [x^n] 1/((1 - n*x) * Product_{k=0..n-1} (1 - k*x)^2).

1, 1, 11, 222, 6627, 262570, 12978758, 769079444, 53138842515, 4194648739710, 372421403333850, 36733739199892020, 3985122473105099406, 471598870326072262644, 60456151456891375730860, 8345905345383943433713800, 1234395864446065862689721475, 194738649118647202909304657910

Offset: 0

Author

Seiichi Manyama, May 13 2025

Keywords

• nonn

Crossrefs

Cf. A350376, A383880.

Cf. A187235, A287532.

Programs

Mathematica

Join[{1}, Table[Sum[StirlingS2[n + k - 1, n - 1]*StirlingS2[2*n - k, n], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 14 2025 *)

PARI

a(n) = polcoef(1/((1-n*x)*prod(k=0, n-1, 1-k*x+x*O(x^n))^2), n);

Formula

a(n) = Sum_{k=0..n} Stirling2(n+k-1,n-1) * Stirling2(2*n-k,n) for n > 0.

a(n) = A287532(n,n).

a(n) ~ 3^(3*n - 1/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n + 1/2) * exp(n) * (3 - 2*w)^n * w^(2*n - 1/2)), where w = -LambertW(-3*exp(-3/2)/2). - Vaclav Kotesovec, May 14 2025