%I #9 Dec 16 2019 07:58:51

%S 1,1,0,10,-68,818,-9782,130730,-1835752,27408672,-438578616,

%T 7697802264,-150743052528,3293454634416,-78787556904864,

%U 2014008113598432,-54001416897306240,1504891127666322048,-43527807706621236480,1311515508480252542208

%N Expansion of e.g.f. -Sum_{k>=1} log(1 - log(1 + x)^k) / k.

%H Vaclav Kotesovec, <a href="/A330352/b330352.txt">Table of n, a(n) for n = 1..400</a>

%F E.g.f.: Sum_{i>=1} Sum_{j>=1} log(1 + x)^(i*j) / (i*j).

%F E.g.f.: log(Product_{k>=1} 1 / (1 - log(1 + x)^k)^(1/k)).

%F a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * tau(k), where tau = A000005.

%t nmax = 20; CoefficientList[Series[-Sum[Log[1 - Log[1 + x]^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%t Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]

%Y Cf. A000005, A002744, A008275, A028342, A089064, A318249, A330351, A330353, A330354, A330493.

%K sign

%O 1,4

%A _Ilya Gutkovskiy_, Dec 11 2019