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A358687 a(n) = n! * Sum_{k=0..n} k^(3 * (n-k)) / (n-k)!.

%I A358687 #24 Nov 27 2022 06:44:28
%S A358687 1,1,4,57,1444,61785,4050126,373648513,47101090744,7764843893265,
%T A358687 1630744323319450,426925697290933401,136591846585403311620,
%U A358687 52602030074554601172649,24058544668572618782040022,12916480280574798627072144465
%N A358687 a(n) = n! * Sum_{k=0..n} k^(3 * (n-k)) / (n-k)!.
%H A358687 Seiichi Manyama, <a href="/A358687/b358687.txt">Table of n, a(n) for n = 0..221</a>
%F A358687 E.g.f.: Sum_{k>=0} x^k * exp(k^3 * x).
%F A358687 G.f.: Sum_{k>=0} k! * x^k / (1 - k^3 * x)^(k+1).
%F A358687 log(a(n)) ~ (6*n*(log(n) - 1) + 3*log(1 + LambertW(n^(2/3))) + 2*n*LambertW(n^(2/3)) * (7*log(n) - 6*log(1 + LambertW(n^(2/3))) + 3*LambertW(n^(2/3)))) / (6*(1 + LambertW(n^(2/3)))). - _Vaclav Kotesovec_, Nov 27 2022
%t A358687 Join[{1}, Table[n! * Sum[k^(3*(n-k))/(n-k)!, {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Nov 27 2022 *)
%o A358687 (PARI) a(n) = n!*sum(k=0, n, k^(3*(n-k))/(n-k)!);
%o A358687 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k*exp(x)^k^3)))
%o A358687 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k/(1-k^3*x)^(k+1)))
%Y A358687 Cf. A006153, A193421, A358688.
%Y A358687 Cf. A349880, A356673.
%K A358687 nonn
%O A358687 0,3
%A A358687 _Seiichi Manyama_, Nov 26 2022