login
A263414
Expansion of Product_{k>=1} 1/(1-x^(3*k+4))^k.
6
1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 4, 2, 0, 5, 6, 1, 6, 10, 2, 7, 19, 6, 9, 28, 14, 11, 44, 28, 16, 61, 52, 25, 87, 93, 45, 116, 153, 77, 160, 244, 141, 215, 376, 244, 301, 560, 422, 422, 817, 695, 617, 1173, 1132, 917, 1661, 1776, 1399, 2331
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).
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)).
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)).
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
Cf. A262877, A262876, A263405 (v=1), A263406 (v=2), A263415 (v=5), A263031, A263416.
Sequence in context: A330369 A309577 A029301 * A162934 A303908 A351592
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 17 2015
STATUS
approved