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A263415
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Expansion of Product_{k>=1} 1/(1-x^(3*k+5))^k.
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6
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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, 10, 14, 44, 12, 28, 60, 17, 52, 86, 26, 93, 112, 46, 152, 152, 78, 243, 196, 142, 372, 264, 244, 552, 350, 422, 798, 486, 692, 1136, 680, 1125, 1582, 997, 1758
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OFFSET
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0,12
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COMMENTS
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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)).
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LINKS
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FORMULA
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G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^(3*k))^2).
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...
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MAPLE
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with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
add(add(`if`(irem(d-3, 3, 'r')=2, d*r, 0)
, d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+5))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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