This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A263415 #18 Nov 16 2024 17:32:02
%S A263415 1,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,1,4,0,2,5,0,6,6,1,10,7,2,19,8,6,28,
%T A263415 10,14,44,12,28,60,17,52,86,26,93,112,46,152,152,78,243,196,142,372,
%U A263415 264,244,552,350,422,798,486,692,1136,680,1125,1582,997,1758
%N A263415 Expansion of Product_{k>=1} 1/(1-x^(3*k+5))^k.
%C A263415 In general, if v>0, GCD(v,3)=1 and g.f. = Product_{k>=1} 1/(1-x^(3*k+v))^k, then
%C A263415 a(n) ~ d3(v) * 3^(v^2/27 - 8/9) * exp(-Pi^4 * v^2 / (3888*Zeta(3)) - v * Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) * n^(v^2/54 - 25/36) / (sqrt(Pi) * 2^(v^2/54 + 11/36) * Zeta(3)^(v^2/54 - 7/36)), where
%C A263415 d3(v) = exp(Integral_{x=0..infinity} (exp((3-v)*x) / (exp(3*x)-1)^2 + (1/12 - v^2/18)/exp(x) - 1/(9*x^2) + v/(9*x))/x dx).
%C A263415 if mod(v,3)=1, then d3(v) = exp(A263031) * 2^((v+2)/6) * 3^((v+2)/18) * Pi^((v+2)/6) / (Gamma(1/3)^((v+2)/3) * A263416((v-1)/3)).
%C A263415 if mod(v,3)=2, then d3(v) = exp(A263030) * 2^((v+1)/6) * Pi^((v+1)/6) / (3^((v+1)/18) * Gamma(2/3)^((v+1)/3) * A263417((v-2)/3)).
%H A263415 Vaclav Kotesovec, <a href="/A263415/b263415.txt">Table of n, a(n) for n = 0..10000</a>
%H A263415 Vaclav Kotesovec, <a href="/A263415/a263415.jpg">Graph - The asymptotic ratio (80000 terms)</a>
%F A263415 G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^(3*k))^2)).
%F A263415 a(n) ~ c * 3^(1/27) * exp(-25*Pi^4 / (3888*Zeta(3)) - 5*Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (sqrt(Pi) * 2^(83/108) * Zeta(3)^(29/108) * n^(25/108)), where c = exp(A263030) * Pi / (3^(1/3) * Gamma(2/3)^2) = 0.98365214791227284535715328899346961376609...
%p A263415 with(numtheory):
%p A263415 a:= proc(n) option remember; local r; `if`(n=0, 1,
%p A263415 add(add(`if`(irem(d-3, 3, 'r')=2, d*r, 0)
%p A263415 , d=divisors(j))*a(n-j), j=1..n)/n)
%p A263415 end:
%p A263415 seq(a(n), n=0..70); # _Alois P. Heinz_, Oct 17 2015
%t A263415 nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+5))^k,{k,1,nmax}],{x,0,nmax}],x]
%t A263415 nmax = 80; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263415 Cf. A262877, A262876, A263405 (v=1), A263406 (v=2), A263414 (v=4), A263030, A263417.
%K A263415 nonn
%O A263415 0,12
%A A263415 _Vaclav Kotesovec_, Oct 17 2015