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A280938
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Expansion of Product_{k>=1} (1 - x^(8*(2*k-1))) * (1 - x^(8*k)) / (1 - x^k).
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6
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1, 1, 2, 3, 5, 7, 11, 15, 20, 28, 38, 50, 67, 87, 113, 146, 187, 237, 301, 378, 473, 590, 732, 903, 1113, 1364, 1666, 2030, 2464, 2981, 3600, 4332, 5201, 6229, 7442, 8869, 10551, 12521, 14829, 17531, 20684, 24357, 28638, 33607, 39375, 46062, 53798, 62736
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OFFSET
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0,3
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COMMENTS
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In general, if r>=2 and g.f. = Product_{k>=1} (1-x^(r*(2*k-1))) * (1-x^(r*k)) / (1-x^k), then
a(n, r) ~ 2*Pi * BesselI(1, Pi/6 * sqrt((24*n-1)*(2*r-3)/(2*r))) / (r*sqrt((24*n-1)/(2*r-3))).
a(n, r) ~ exp(Pi * sqrt((2/3 - 1/r)*n)) * (2*r-3)^(1/4) / (2 * 3^(1/4) * r^(3/4) * n^(3/4)) * (1 -(3*sqrt(3*r)/(8*Pi*sqrt(2*r-3)) + Pi*sqrt(2*r-3)/(48*sqrt(3*r))) / sqrt(n) + (Pi^2*(2*r-3)/(13824*r) - 45*r/(128*Pi^2*(2*r-3)) + 5/128)/n).
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REFERENCES
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D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.
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LINKS
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FORMULA
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a(n) ~ Pi * BesselI(1, Pi * sqrt(13*(24*n-1))/24) / (4*sqrt((24*n-1)/13)).
a(n) ~ exp(Pi*sqrt(13*n/6)/2) * 13^(1/4) / (2^(13/4) * 3^(1/4) * n^(3/4)) * (1 -(3*sqrt(3)/(2*Pi*sqrt(26)) + Pi*sqrt(13)/(96*sqrt(6)))/sqrt(n) + (13*Pi^2/110592 - 45/(208*Pi^2) + 5/128)/n).
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[(1-x^(8*(2*k-1))) * (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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nonn
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AUTHOR
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STATUS
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approved
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