OFFSET
0,11
COMMENTS
In general, if v>0, GCD(v,3)=1 and g.f. = Product_{k>=1} 1/(1-x^(3*k+v))^k, then
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
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).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (80000 terms)
FORMULA
G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^(3*k))^2).
a(n) ~ c * exp(-Pi^4/(243*Zeta(3)) - 4*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^(65/108) * 3^(8/27) * Zeta(3)^(11/108) * n^(43/108)), where c = exp(A263031) * 2 * 3^(1/3) * Pi / Gamma(1/3)^2 = 1.24446091929106216111829684663735422946506...
MAPLE
with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
add(add(`if`(irem(d-3, 3, 'r')=1, d*r, 0)
, d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..70); # Alois P. Heinz, Oct 17 2015
MATHEMATICA
nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+4))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 17 2015
STATUS
approved