A353252 Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j + 2 * x).

1, 0, 2, 2, 8, 24, 100, 488, 2832, 19096, 147296, 1281392, 12422864, 132870368, 1554525152, 19750621216, 270817685568, 3986140113792, 62686410981696, 1048946532137216, 18608550117641728, 348854564104019072, 6891109834644748032, 143058034748452036352

Offset: 0

Author

Seiichi Manyama, Apr 08 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..451

Crossrefs

Cf. A352802, A343579, A353253, A353254.

Cf. A097341.

Programs

Mathematica

a[n_] := Sum[2^k * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+2*x)))

PARI

a(n) = sum(k=0, n\2, 2^k*abs(stirling(n-k, k, 1)));

Formula

a(n) = Sum_{k=0..floor(n/2)} 2^k * |Stirling1(n-k,k)|.