A330354 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)).

1, 2, 1, 21, -122, 1752, -21730, 309166, -4521032, 70344768, -1173530712, 21642745704, -448130571696, 10352684535840, -260101132095888, 6921279885508848, -191813249398678272, 5502934340821289088, -163695952380982280832, 5078687529186002247552

Offset

1,2

Links

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

Formula

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

E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306042.

exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of the partition numbers (A000041).

a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * sigma(k), where sigma = A000203.

Conjecture: a(n) ~ n! * (-1)^n * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Dec 16 2019

Mathematica

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

Crossrefs

Cf. A000041, A000203, A002743, A008275, A038048, A089064, A306042, A330351, A330352, A330353, A330494.

Sequence in context: A164827 A345760 A213976 * A200859 A127607 A255861

Adjacent sequences: A330351 A330352 A330353 * A330355 A330356 A330357

Keyword

sign

Author

Ilya Gutkovskiy, Dec 11 2019

Status

approved