A337001 a(n) = n! * Sum_{k=0..n} k^3 / k!.

0, 1, 10, 57, 292, 1585, 9726, 68425, 547912, 4931937, 49320370, 542525401, 6510306540, 84633987217, 1184875823782, 17773137360105, 284370197765776, 4834293362023105, 87017280516421722, 1653328329812019577, 33066566596240399540, 694397898521048399601

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2020

Keywords

• nonn

Comments

Exponential convolution of cubes (A000578) and factorial numbers (A000142).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A000142, A000522, A000578, A007526, A030297, A256016, A337002.

Row sums of A121682.

Programs

Mathematica

Table[n! Sum[k^3/k!, {k, 0, n}], {n, 0, 21}]
nmax = 21; CoefficientList[Series[x (1 + 3 x + x^2) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = n (n^2 + a[n - 1]); Table[a[n], {n, 0, 21}]

PARI

a(n) = n! * sum(k=0, n, k^3/k!); \\ Michel Marcus, Aug 12 2020

Formula

E.g.f.: x * (1 + 3*x + x^2) * exp(x) / (1 - x).

a(0) = 0; a(n) = n * (n^2 + a(n-1)).

a(n) ~ 5*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024