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%I #19 Feb 04 2022 08:22:45
%S 1,2,30,1108,76372,8463328,1375868768,308440047648,91189383264864,
%T 34376022491122368,16093445542120281792,9160424435706947112576,
%U 6230035512106223752576896,4989402076922846372194268160,4647526704475074504983564884992
%N a(n) = Sum_{k=0..n} k! * 2^k * k^n * Stirling1(n,k).
%H Seiichi Manyama, <a href="/A350719/b350719.txt">Table of n, a(n) for n = 0..221</a>
%F E.g.f.: Sum_{k>=0} (2 * log(1 + k*x))^k.
%t a[0] = 1; a[n_] := Sum[k! * 2^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 15, 0] (* _Amiram Eldar_, Feb 03 2022 *)
%o (PARI) a(n) = sum(k=0, n, k!*2^k*k^n*stirling(n, k, 1));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*log(1+k*x))^k)))
%Y Cf. A320083, A350720.
%Y Cf. A088501, A195005, A350721.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Feb 03 2022