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 A306038 #9 Feb 16 2025 08:33:54
%S A306038 1,2,3,5,12,34,122,482,2328,11640,71952,424368,3312240,21357504,
%T A306038 217045488,1351338864,19990187520,89379824256,2631270916224,
%U A306038 892036259712,507945420198144,-3068802187635456,142961233091051520,-1849617314640322560,55640352746480440320
%N A306038 Expansion of e.g.f. (1 + x)/(1 - log(1 + x)).
%H A306038 Robert Israel, <a href="/A306038/b306038.txt">Table of n, a(n) for n = 0..452</a>
%H A306038 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A306038 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A306038 a(n) = Sum_{k=0..n} Stirling1(n,k)*A000522(k).
%e A306038 (1 + x)/(1 - log(1 + x)) = 1 + 2*x + 3*x^2/2! + 5*x^3/3! + 12*x^4/4! + 34*x^5/5! + 122*x^6/6! + ...
%p A306038 S:= series((1+x)/(1-log(1+x)),x,51):
%p A306038 seq(coeff(S,x,j)*j!,j=0..50); # _Robert Israel_, Jun 19 2018
%t A306038 nmax = 24; CoefficientList[Series[(1 + x)/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A306038 FullSimplify[Table[Sum[StirlingS1[n, k] E Gamma[1 + k, 1], {k, 0, n}], {n, 0, 24}]]
%Y A306038 Cf. A000522, A006252, A059099, A089064.
%K A306038 sign
%O A306038 0,2
%A A306038 _Ilya Gutkovskiy_, Jun 17 2018