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A263364 Expansion of Product_{k>=1} 1/(1-x^(k+8))^k. 8
1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 18, 23, 33, 43, 60, 77, 103, 130, 168, 209, 267, 331, 420, 526, 667, 839, 1069, 1347, 1711, 2160, 2733, 3437, 4336, 5435, 6828, 8543, 10699, 13357, 16703, 20820, 25986, 32362, 40327, 50152, 62407 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.

d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).

d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).

d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

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

Eric Weisstein's World of Mathematics, Barnes G-Function.

FORMULA

G.f.: exp(Sum_{k>=1} x^(9*k)/(k*(1-x^k)^2).

a(n) ~ exp(1/12 - 4*Pi^4/(27*Zeta(3)) - 2^(5/3) * Pi^2 * n^(1/3) / (3 * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(359/36) * Pi^(7/2) / (8026324992000 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(377/36)), 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-8), d=divisors(j))*a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2015

MATHEMATICA

nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+8))^k, {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 60; CoefficientList[Series[E^Sum[x^(9*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A000219 (v=0), A052847 (v=1), A263358 (v=2), A263359 (v=3), A263360 (v=4), A263361 (v=5), A263362 (v=6), A263363 (v=7).

Sequence in context: A122427 A161597 A320320 * A296242 A331020 A085429

Adjacent sequences:  A263361 A263362 A263363 * A263365 A263366 A263367

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 16 2015

STATUS

approved

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Last modified August 4 10:06 EDT 2021. Contains 346447 sequences. (Running on oeis4.)