%I #9 Mar 27 2019 03:53:36

%S 1,1,2,7,31,178,1200,9588,86592,887086,10035164,125472246,1705102394,

%T 25175822644,399387494956,6801042408728,123348694663480,

%U 2379855020533664,48569042602254128,1047134236970183664,23748242269316806752,565834452464428045872,14117321495269290091440

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

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>

%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A007840(k).

%F a(n) ~ n! * exp(-exp(-1)) / (exp(1 - exp(-1)) - 1)^(n+1). - _Vaclav Kotesovec_, Jul 01 2018

%e 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 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 nmax = 22; CoefficientList[Series[1/(1 + Log[1 - Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Sum[StirlingS1[n, k] Abs[StirlingS1[k, j]] j!, {j, 0, k}], {k, 0, n}], {n, 0, 22}]

%t 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 Cf. A007840, A089064, A305323, A305988.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jun 17 2018