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A261733
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Expansion of Product_{k>=1} (1 + x^(9*k))/(1 + x^k).
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11
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1, -1, 0, -1, 1, -1, 1, -1, 2, -1, 1, -2, 2, -2, 2, -3, 4, -3, 4, -5, 5, -6, 6, -7, 8, -8, 9, -9, 10, -12, 11, -13, 15, -16, 17, -18, 22, -23, 23, -27, 30, -31, 32, -35, 40, -40, 42, -48, 51, -54, 57, -63, 69, -71, 78, -85, 90, -97, 102, -110, 118, -124, 133
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OFFSET
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0,9
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COMMENTS
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In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd.
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LINKS
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FORMULA
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a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)/3) / (2 * 3^(3/4) * n^(3/4)).
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, -1, 0,
-1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1]
[1+irem(d, 18)], d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[(1 + x^(9*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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