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A305628 Expansion of Product_{k>=1} 1/(1 + x^k)^(k+1). 2
1, -2, 0, -2, 5, -2, 7, -6, 11, -20, 13, -32, 31, -50, 60, -70, 124, -112, 192, -198, 295, -364, 422, -616, 661, -1002, 1034, -1500, 1737, -2208, 2808, -3234, 4462, -4876, 6735, -7464, 9990, -11610, 14410, -17866, 20947, -27082, 30493, -40056, 45147, -58196, 66999, -83278, 99641 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Convolution of A081362 and A255528.

Convolution inverse of A219555.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

FORMULA

G.f.: exp(Sum_{k>=1} (-1)^k*x^k*(2 - x^k)/(k*(1 - x^k)^2)).

a(n) ~ (-1)^n * exp(3 * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (3 * 2^(7/3) * Zeta(3)^(1/3)) - 1/12 - Pi^4 / (864 * Zeta(3))) * A * Zeta(3)^(5/36) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Aug 21 2018

MATHEMATICA

nmax = 48; CoefficientList[Series[Product[1/(1 + x^k)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 48; CoefficientList[Series[Exp[Sum[(-1)^k x^k (2 - x^k)/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d (d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

CROSSREFS

Cf. A081362, A219555, A255528.

Sequence in context: A161564 A078182 A133394 * A094721 A301951 A144529

Adjacent sequences:  A305625 A305626 A305627 * A305629 A305630 A305631

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Aug 11 2018

STATUS

approved

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Last modified July 24 20:26 EDT 2021. Contains 346273 sequences. (Running on oeis4.)