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A266821
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Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).
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4
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1, 4, 8, 24, 44, 88, 176, 312, 544, 924, 1584, 2552, 4136, 6488, 10128, 15632, 23748, 35640, 53080, 78136, 114024, 165552, 237744, 339544, 481248, 678236, 949008, 1321840, 1830376, 2521688, 3456672, 4717208, 6406680, 8666448, 11672464, 15660528, 20934868
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OFFSET
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0,2
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COMMENTS
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In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 - x^k)) then a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(m+1)*n), where c = 2*Pi^2/3 + log(m)^2 + 2*polylog(2, -1/m).
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LINKS
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FORMULA
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a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (8*Pi*n), where c = 2*Pi^2/3 + log(3)^2 + 2*polylog(2, -1/3) = 7.16861897522987077909937377164783326088308015803... .
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(t-> b(t, min(t, i-1)))(n-i*j), j=1..n/i)*4 +b(n, i-1)))
end:
a:= n-> b(n$2):
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[(1+3*x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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PROG
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(PARI) { my(n=40); Vec(prod(k=1, n, 4/(1-x^k) - 3 + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017
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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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