A320350 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k).

1, 1, 3, 20, 148, 1384, 15728, 207696, 3094152, 51423288, 945943512, 19083180192, 418550811600, 9907493349168, 251588827187280, 6820899616891008, 196645361557479552, 6007407711127690752, 193842462200078260224, 6586904673329133618432, 235079477736802622742528, 8790132360155070084076800

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000009(k)*k!.

From Vaclav Kotesovec, Oct 13 2018: (Start)

a(n) ~ n! * exp(n + Pi*sqrt(n/(3*(exp(1) - 1))) + Pi^2/(24*(exp(1) - 1))) / (4 * 3^(1/4) * n^(3/4) * (exp(1) - 1)^(n + 1/4)).

a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(3*(exp(1) - 1))) + Pi^2/(24*(exp(1) - 1))) * n^(n - 1/4) / (2^(3/2) * 3^(1/4) * (exp(1) - 1)^(n + 1/4)).

(End)

Maple

seq(n!*coeff(series(mul((1 + log(1/(1 - x))^k), k=1..100), x=0, 22), x, n), n=0..21); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 21; CoefficientList[Series[Product[(1 + Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]

Crossrefs

Cf. A000009, A048994, A088311, A298905, A305550, A320349.

Sequence in context: A074569 A026303 A154627 * A354495 A336283 A091172

Adjacent sequences: A320347 A320348 A320349 * A320351 A320352 A320353

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 11 2018

Status

approved