A263364 - OEIS

%I #17 Nov 16 2024 16:10:51

%S 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,

%T 168,209,267,331,420,526,667,839,1069,1347,1711,2160,2733,3437,4336,

%U 5435,6828,8543,10699,13357,16703,20820,25986,32362,40327,50152,62407

%N Expansion of Product_{k>=1} 1/(1-x^(k+8))^k.

%C 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.

%C 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).

%C 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).

%C 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.

%H Vaclav Kotesovec, <a href="/A263364/b263364.txt">Table of n, a(n) for n = 0..5000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.

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

%F 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.

%p with(numtheory):

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

%p       max(0, d-8), d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 16 2015

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

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

%Y 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).

%K nonn,changed

%O 0,11

%A _Vaclav Kotesovec_, Oct 16 2015