A280938 Expansion of Product_{k>=1} (1 - x^(8*(2*k-1))) * (1 - x^(8*k)) / (1 - x^k).

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

Offset: 0

Author

Vaclav Kotesovec, Jan 11 2017

Keywords

• nonn

Comments

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).

References

• D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), no. 227, 54 pp.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
• Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
• Wikipedia, Bailey pair.

Crossrefs

Cf. A000700 (r=2), A070047 (r=3), A108961 (r=4), A108962 (r=5), A271661 (r=6), A280937 (r=7).

Programs

Mathematica

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]

Formula

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).