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%I #12 Jan 13 2024 06:51:01
%S 0,1,18,135,796,4605,28926,204883,1643160,14795001,147960010,
%T 1627574751,19530917748,253901959285,3554627468406,53319412076715,
%U 853110593292976,14502880086064113,261051841549259010,4959984989436051511,99199699788721190220
%N a(n) = n! * Sum_{k=0..n} k^4 / k!.
%C Exponential convolution of fourth powers (A000583) and factorial numbers (A000142).
%H Seiichi Manyama, <a href="/A337002/b337002.txt">Table of n, a(n) for n = 0..449</a>
%F E.g.f.: x * (1 + 7*x + 6*x^2 + x^3) * exp(x) / (1 - x).
%F a(0) = 0; a(n) = n * (n^3 + a(n-1)).
%F a(n) ~ 15*exp(1)*n!. - _Vaclav Kotesovec_, Jan 13 2024
%t Table[n! Sum[k^4/k!, {k, 0, n}], {n, 0, 20}]
%t nmax = 20; CoefficientList[Series[x (1 + 7 x + 6 x^2 + x^3) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 0; a[n_] := a[n] = n (n^3 + a[n - 1]); Table[a[n], {n, 0, 20}]
%o (PARI) a(n) = n! * sum(k=0, n, k^4/k!); \\ _Michel Marcus_, Aug 12 2020
%Y Cf. A000142, A000522, A000583, A007526, A030297, A256016, A337001.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 10 2020