A307523 - OEIS

%I #7 Apr 13 2019 04:26:02

%S 1,2,6,28,124,848,5312,40080,367632,3132096,27731328,474979008,

%T 1130161728,90279554688,268809015168,3005011325952,473192066191104,

%U -7913323872693504,186235895195313408,1357401816746159616,-181477915903332002304,9552839425392612096000

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

%C Exponential convolution of A298905 and A306042.

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

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

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

%t 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]!

%t 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]!

%t nmax = 21; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[StirlingS1[n, k] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 21}]

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

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Apr 12 2019