A330352 Expansion of e.g.f. -Sum_{k>=1} log(1 - log(1 + x)^k) / k.

1, 1, 0, 10, -68, 818, -9782, 130730, -1835752, 27408672, -438578616, 7697802264, -150743052528, 3293454634416, -78787556904864, 2014008113598432, -54001416897306240, 1504891127666322048, -43527807706621236480, 1311515508480252542208

Offset: 1

Author

Ilya Gutkovskiy, Dec 11 2019

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Crossrefs

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

Programs

Mathematica

nmax = 20; CoefficientList[Series[-Sum[Log[1 - Log[1 + x]^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]

Formula

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

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

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