%I #27 Feb 04 2022 09:25:10
%S 1,1,129,121172,421875178,3922823960054,80130334773241142,
%T 3156849112458066440568,218554371053209725986724984,
%U 24795129220015277612148345850896,4365539219231132131300647267518575008,1141930521329052244894253748456776246166288
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3*n) * Stirling1(n,k).
%C In general, for m >= 0, Sum_{k=0..n} (-1)^(n-k) * k! * k^(m*n) * Stirling1(n,k) ~ c * r^(m*n) * (1 + r*exp(m/r))^n * n^((m+1)*n + 1/2) / exp((m+1)*n), where r is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-m/r) and c is a constant (depending only on m). - _Vaclav Kotesovec_, Feb 04 2022
%H Seiichi Manyama, <a href="/A351137/b351137.txt">Table of n, a(n) for n = 0..125</a>
%F E.g.f.: Sum_{k>=0} (-log(1 - k^3*x))^k.
%F a(n) ~ c * r^(3*n) * (1 + r*exp(3/r))^n * n^(4*n + 1/2) / exp(4*n), where r = 0.97698437755148201976772582981871258235824532360125531194... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-3/r) and c = 2.3655154360078103511101518906595610482889989819... - _Vaclav Kotesovec_, Feb 04 2022
%t a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(3*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 12, 0] (* _Amiram Eldar_, Feb 02 2022 *)
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^(3*n)*stirling(n, k, 1));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^3*x))^k)))
%Y Cf. A007840 (m=0), A320096 (m=1), A351136 (m=2).
%Y Cf. A203798, A351134, A351138.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Feb 02 2022