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A306037 Expansion of e.g.f. 1/(1 + log(1 - log(1 + x))).

%I A306037 #10 Feb 16 2025 08:33:54
%S A306037 1,1,2,7,31,178,1200,9588,86592,887086,10035164,125472246,1705102394,
%T A306037 25175822644,399387494956,6801042408728,123348694663480,
%U A306037 2379855020533664,48569042602254128,1047134236970183664,23748242269316806752,565834452464428045872,14117321495269290091440
%N A306037 Expansion of e.g.f. 1/(1 + log(1 - log(1 + x))).
%H A306037 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A306037 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A306037 a(n) = Sum_{k=0..n} Stirling1(n,k)*A007840(k).
%F A306037 a(n) ~ n! * exp(-exp(-1)) / (exp(1 - exp(-1)) - 1)^(n+1). - _Vaclav Kotesovec_, Jul 01 2018
%e A306037 1/(1 + log(1 - log(1 + x))) = 1 + x + 2*x^2/2! + 7*x^3/3! + 31*x^4/4! + 178*x^5/5! + 1200*x^6/6! + ...
%p A306037 a:=series(1/(1+log(1-log(1+x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 26 2019
%t A306037 nmax = 22; CoefficientList[Series[1/(1 + Log[1 - Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
%t A306037 Table[Sum[Sum[StirlingS1[n, k] Abs[StirlingS1[k, j]] j!, {j, 0, k}], {k, 0, n}], {n, 0, 22}]
%t A306037 a[0] = 1; a[n_] := a[n] = Sum[Sum[(j - 1)! StirlingS1[k, j], {j, 1, k}] a[n - k]/k!, {k, 1, n}]; Table[n! a[n], {n, 0, 22}]
%Y A306037 Cf. A007840, A089064, A305323, A305988.
%K A306037 nonn
%O A306037 0,3
%A A306037 _Ilya Gutkovskiy_, Jun 17 2018