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A261735
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Expansion of Product_{k>=1} (1 + x^(8*k))/(1 + x^k).
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11
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1, -1, 0, -1, 1, -1, 1, -1, 3, -3, 2, -3, 4, -4, 4, -5, 8, -8, 7, -9, 11, -12, 12, -14, 20, -21, 19, -24, 28, -30, 31, -35, 45, -48, 47, -55, 64, -68, 71, -80, 97, -103, 104, -119, 135, -145, 152, -168, 198, -211, 216, -243, 272, -291, 307, -337, 386, -412
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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(sqrt(5*n/6)*Pi/2) * 5^(1/4) / (2^(11/4)*3^(1/4)*n^(3/4)).
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MAPLE
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seq(coeff(series(mul((1+x^(8*k))/(1+x^k), k=1..n), x, n+1), x, n), n=0..60); # Muniru A Asiru, Jul 29 2018
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[(1 + x^(8*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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