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A307524 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k)/(1 - log(1/(1 - x))^k). 1
1, 2, 10, 76, 724, 8368, 113792, 1771824, 31001424, 601677888, 12818974848, 297223165248, 7446226027584, 200354793323904, 5760239869401984, 176170480317568512, 5709535272618925824, 195419487662892221184, 7042458625343222876928, 266500916470984705887744 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Exponential convolution of A320349 and A320350.

LINKS

Table of n, a(n) for 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

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}]

CROSSREFS

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

Sequence in context: A321394 A320956 A324061 * A066223 A088500 A295929

Adjacent sequences:  A307521 A307522 A307523 * A307525 A307526 A307527

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 12 2019

STATUS

approved

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Last modified January 19 00:40 EST 2020. Contains 331030 sequences. (Running on oeis4.)