A383052 a(n) = Sum_{k=0..n} (k+1)^3 * Stirling2(n,k).

1, 8, 35, 153, 706, 3479, 18313, 102678, 610989, 3844525, 25492752, 177579961, 1295811637, 9879799744, 78525094847, 649253421173, 5573667453498, 49595062947091, 456689512735421, 4345710521536150, 42675672248378721, 431963852263306569, 4501627598926298992

Offset: 0

Author

Seiichi Manyama, Apr 14 2025

Keywords

• nonn

Comments

Stirling transform of (n+1)^3.

Links

• Eric Weisstein's World of Mathematics, Stirling Transform

Crossrefs

Cf. A000110, A222636, A362925.

Programs

PARI

a(n) = sum(k=0, n, (k+1)^3*stirling(n, k, 2));

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^3*(exp(x)-1)^k/k!)))

Formula

a(n) = A362925(n+3,3).

E.g.f.: Sum_{k>=0} (k+1)^3 * (exp(x) - 1)^k / k!.

E.g.f.: exp(exp(x) - 1) * Sum_{k=0..3} Stirling2(4,k+1) * (exp(x) - 1)^k.