A320349 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k).

1, 1, 5, 32, 278, 2894, 35986, 514128, 8306448, 149558688, 2968216944, 64314676128, 1510065781968, 38178537908016, 1033794746169168, 29840453678758272, 914461132860063360, 29645845798652997120, 1013511411165693991680, 36436289007997132646400, 1373976152501162688288000

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..413

Formula

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

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000041(k)*k!.

From Vaclav Kotesovec, Oct 13 2018: (Start)

a(n) ~ n! * exp(n + Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) / (4 * sqrt(3) * n * (exp(1) - 1)^n).

a(n) ~ sqrt(Pi) * exp(Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) * n^(n - 1/2) / (2^(3/2) * sqrt(3) * (exp(1) - 1)^n).

(End)

Maple

seq(n!*coeff(series(mul(1/(1-log(1/(1-x))^k), k=1..100), x=0, 21), x, n), n=0..20); # Paolo P. Lava, Jan 09 2019

Mathematica

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

Crossrefs

Cf. A000041, A000203, A048994, A053529, A167137, A306042, A320350.

Sequence in context: A328055 A265130 A305407 * A354013 A331660 A001923

Adjacent sequences: A320346 A320347 A320348 * A320350 A320351 A320352

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 11 2018

Status

approved