A337002 a(n) = n! * Sum_{k=0..n} k^4 / k!.

0, 1, 18, 135, 796, 4605, 28926, 204883, 1643160, 14795001, 147960010, 1627574751, 19530917748, 253901959285, 3554627468406, 53319412076715, 853110593292976, 14502880086064113, 261051841549259010, 4959984989436051511, 99199699788721190220

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2020

Keywords

• nonn

Comments

Exponential convolution of fourth powers (A000583) and factorial numbers (A000142).

Links

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

Crossrefs

Cf. A000142, A000522, A000583, A007526, A030297, A256016, A337001.

Programs

Mathematica

Table[n! Sum[k^4/k!, {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[x (1 + 7 x + 6 x^2 + x^3) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = n (n^3 + a[n - 1]); Table[a[n], {n, 0, 20}]

PARI

a(n) = n! * sum(k=0, n, k^4/k!); \\ Michel Marcus, Aug 12 2020

Formula

E.g.f.: x * (1 + 7*x + 6*x^2 + x^3) * exp(x) / (1 - x).

a(0) = 0; a(n) = n * (n^3 + a(n-1)).

a(n) ~ 15*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024