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A307523 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k)/(1 - log(1 + x)^k). 1
1, 2, 6, 28, 124, 848, 5312, 40080, 367632, 3132096, 27731328, 474979008, 1130161728, 90279554688, 268809015168, 3005011325952, 473192066191104, -7913323872693504, 186235895195313408, 1357401816746159616, -181477915903332002304, 9552839425392612096000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Exponential convolution of A298905 and A306042.

LINKS

Table of n, a(n) for 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!.

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}]

CROSSREFS

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

Sequence in context: A047125 A189238 A226497 * A065577 A227294 A302336

Adjacent sequences:  A307520 A307521 A307522 * A307524 A307525 A307526

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Apr 12 2019

STATUS

approved

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Last modified January 19 17:59 EST 2020. Contains 331051 sequences. (Running on oeis4.)