A307524 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k)/(1 - log(1/(1 - x))^k).

1, 2, 10, 76, 724, 8368, 113792, 1771824, 31001424, 601677888, 12818974848, 297223165248, 7446226027584, 200354793323904, 5760239869401984, 176170480317568512, 5709535272618925824, 195419487662892221184, 7042458625343222876928, 266500916470984705887744

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2019

Keywords

• nonn

Comments

Exponential convolution of A320349 and A320350.

Crossrefs

Cf. A015128, A306045, A307523, A320349, A320350.

Programs

Mathematica

nmax = 19; CoefficientList[Series[Product[(1 + Log[1/(1 - x)]^k)/(1 - Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) Log[1/(1 - x)]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1/(1 - x)]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 19}]

Formula

E.g.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*log(1/(1 - x))^k/k).

E.g.f.: 1/theta_4(log(1/(1 - x))).

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A015128(k)*k!.

a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(exp(1)-1)) + Pi^2/(8*(exp(1)-1))) * n^(n - 1/2) / (2^(5/2) * (exp(1)-1)^n). - Vaclav Kotesovec, Apr 13 2019