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A263360 Expansion of Product_{k>=1} 1/(1-x^(k+4))^k. 8
1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 14, 19, 29, 40, 58, 79, 113, 153, 215, 294, 407, 555, 767, 1040, 1424, 1930, 2624, 3540, 4794, 6441, 8677, 11627, 15589, 20818, 27812, 37011, 49257, 65360, 86681, 114665, 151594, 199947, 263530, 346647, 455553, 597628 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

FORMULA

G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1-x^k)^2).

a(n) ~ exp(1/12 - Pi^4/(27*Zeta(3)) - 2^(2/3) * Pi^2 * n^(1/3) / (3 * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(71/36) * Pi^(3/2) / (12 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(89/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

MAPLE

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

      max(0, d-4), d=divisors(j))*a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Oct 16 2015

MATHEMATICA

nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+4))^k, {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 50; CoefficientList[Series[E^Sum[x^(5*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A000219, A052847, A263358, A263359, A263361, A263362, A263363, A263364.

Sequence in context: A286225 A239048 A219898 * A050763 A252023 A282895

Adjacent sequences:  A263357 A263358 A263359 * A263361 A263362 A263363

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 16 2015

STATUS

approved

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Last modified November 24 07:33 EST 2020. Contains 338607 sequences. (Running on oeis4.)