A330450 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)^2).

1, 4, 7, 55, -162, 4100, -49030, 779914, -11928008, 198650880, -3538477560, 70414760136, -1571134087824, 38788172175072, -1028732373217200, 28631225505910224, -826097667884640768, 24664145505337921920, -765245501125015575168, 24841409653689047496576

Offset: 1

Author

Ilya Gutkovskiy, Dec 15 2019

Keywords

• sign

Links

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

Crossrefs

Cf. A000219, A001157, A008275, A318250, A330354, A330449, A330495.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]

Formula

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

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

exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of A000219.

a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * sigma_2(k), where sigma_2 = A001157.

Conjecture: a(n) ~ n! * (-1)^n * zeta(3) * n * exp(n) / (8 * (exp(1) - 1)^(n+2)). - Vaclav Kotesovec, Dec 16 2019