%I #6 Mar 27 2019 03:53:48
%S 1,1,0,3,-14,80,-479,3024,-19802,129114,-740172,1569150,66402733,
%T -2071117841,47637088903,-1025499372275,21965630892954,
%U -479007175968646,10740252324008140,-248679637522491506,5956075167341406752,-147619197124939380988,3784960110596636657926
%N Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k/k!).
%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 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*log(1 + x)^(j*k)/(k*(j!)^k)).
%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A007837(k).
%p a:=series(mul(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 nmax = 22; CoefficientList[Series[Product[(1 + Log[1 + x]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) Log[1 + x]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, -# (-#!)^(-k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS1[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]
%Y Cf. A007837, A305547, A306039.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Jun 17 2018