This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306039 #8 Feb 16 2025 08:33:54
%S A306039 1,1,2,3,14,0,359,-1988,28706,-312210,4387572,-62769366,1006242599,
%T A306039 -17203315363,318393704043,-6296931104285,133039045075494,
%U A306039 -2986262905171914,71018001954178952,-1783064497977512206,47133484019671647932,-1308274154275749372040,38042727898691562357962
%N A306039 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k/k!).
%H A306039 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A306039 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A306039 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} log(1 + x)^(j*k)/(k*(j!)^k)).
%F A306039 a(n) = Sum_{k=0..n} Stirling1(n,k)*A005651(k).
%p A306039 a:=series(mul(1/(1-log(1+x)^k/k!),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 26 2019
%t A306039 nmax = 22; CoefficientList[Series[Product[1/(1 - Log[1 + x]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A306039 nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Log[1 + x]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A306039 Table[Sum[StirlingS1[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]
%Y A306039 Cf. A005651, A140585, A306040.
%K A306039 sign
%O A306039 0,3
%A A306039 _Ilya Gutkovskiy_, Jun 17 2018