Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #18 Feb 04 2022 08:22:49
%S 1,3,69,3948,422082,72567522,18304992558,6367730357160,
%T 2921446409138136,1709074810258369776,1241694104839498851552,
%U 1096850187800368469477424,1157691464039682741551221152,1438880771284303822650674399664
%N a(n) = Sum_{k=0..n} k! * 3^k * k^n * Stirling1(n,k).
%H Seiichi Manyama, <a href="/A350720/b350720.txt">Table of n, a(n) for n = 0..213</a>
%F E.g.f.: Sum_{k>=0} (3 * log(1 + k*x))^k.
%t a[0] = 1; a[n_] := Sum[k! * 3^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* _Amiram Eldar_, Feb 03 2022 *)
%o (PARI) a(n) = sum(k=0, n, k!*3^k*k^n*stirling(n, k, 1));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*log(1+k*x))^k)))
%Y Cf. A320083, A350719.
%Y Cf. A195263, A335531, A350721.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Feb 03 2022