A307523 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k)/(1 - log(1 + x)^k).

1, 2, 6, 28, 124, 848, 5312, 40080, 367632, 3132096, 27731328, 474979008, 1130161728, 90279554688, 268809015168, 3005011325952, 473192066191104, -7913323872693504, 186235895195313408, 1357401816746159616, -181477915903332002304, 9552839425392612096000

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2019

Keywords

• sign

Comments

Exponential convolution of A298905 and A306042.

Crossrefs

Cf. A015128, A298905, A306042, A306045, A307524.

Programs

Mathematica

nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k)/(1 - Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) Log[1 + x]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 21}]

Formula

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

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

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