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A263395
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Expansion of Product_{k>=1} 1/(1 - x^(2*k+5))^k.
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5
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1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 5, 2, 6, 6, 7, 10, 9, 19, 11, 28, 16, 44, 25, 61, 40, 87, 65, 116, 107, 160, 168, 215, 260, 295, 393, 407, 578, 573, 836, 814, 1193, 1167, 1675, 1684, 2335, 2427, 3238, 3501, 4468, 5014, 6161, 7152, 8494, 10121
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OFFSET
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0,10
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LINKS
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FORMULA
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G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^(2*k))^2).
a(n) ~ 2^(109/72) * exp(-1/24 - 25*Pi^4/(1728*Zeta(3)) - 5*Pi^2 * n^(1/3) / (12 * 2^(2/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(25/72) / (3*sqrt(3*Pi) * Zeta(3)^(61/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
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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-4, 2, 'r')=1, d*r, 0)
, d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+5))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^(2*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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