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A263359 Expansion of Product_{k>=1} 1/(1-x^(k+3))^k. 8
1, 0, 0, 0, 1, 2, 3, 4, 6, 8, 13, 18, 29, 40, 61, 86, 127, 178, 260, 364, 524, 734, 1042, 1454, 2051, 2848, 3981, 5510, 7652, 10542, 14558, 19970, 27428, 37480, 51222, 69720, 94870, 128634, 174306, 235506, 317899, 428018, 575688, 772540, 1035538, 1385264 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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^(4*k)/(k*(1-x^k)^2).

a(n) ~ exp(1/12 - Pi^4/(48*Zeta(3)) - Pi^2 * n^(1/3) / (2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(29/36) * Pi / (A * 2^(47/36) * sqrt(3) * Zeta(3)^(47/36)), 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-3), 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+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 50; CoefficientList[Series[E^Sum[x^(4*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A000219, A052847, A263358, A263360, A263361, A263362, A263363, A263364.

Sequence in context: A094372 A039880 A240452 * A246905 A000029 A155051

Adjacent sequences:  A263356 A263357 A263358 * A263360 A263361 A263362

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 16 2015

STATUS

approved

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Last modified October 18 20:10 EDT 2018. Contains 316325 sequences. (Running on oeis4.)