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A263359
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Expansion of Product_{k>=1} 1/(1-x^(k+3))^k.
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8
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1, 0, 0, 0, 1, 2, 3, 4, 6, 8, 13, 18, 29, 40, 61, 86, 127, 178, 260, 364, 524, 734, 1042, 1454, 2051, 2848, 3981, 5510, 7652, 10542, 14558, 19970, 27428, 37480, 51222, 69720, 94870, 128634, 174306, 235506, 317899, 428018, 575688, 772540, 1035538, 1385264
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OFFSET
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0,6
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LINKS
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FORMULA
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G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1-x^k)^2).
a(n) ~ exp(1/12 - Pi^4/(48*Zeta(3)) - Pi^2 * n^(1/3) / (2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(29/36) * Pi / (A * 2^(47/36) * sqrt(3) * Zeta(3)^(47/36)), 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; `if`(n=0, 1, add(add(d*
max(0, d-3), d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[E^Sum[x^(4*k)/(k*(1-x^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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