A280937 Expansion of Product_{k>=1} ((1 - x^(7*(2*k-1))) * (1 - x^(7*k)) / (1 - x^k)).

1, 1, 2, 3, 5, 7, 11, 13, 20, 26, 36, 46, 63, 79, 105, 132, 171, 213, 273, 336, 425, 522, 650, 793, 981, 1188, 1456, 1756, 2136, 2563, 3098, 3698, 4443, 5285, 6312, 7477, 8891, 10489, 12415, 14599, 17206, 20165, 23678, 27659, 32363, 37698, 43958, 51058, 59361

Offset: 0

Author

Vaclav Kotesovec, Jan 11 2017

Keywords

• nonn

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, A070047, A108961, A108962, A271661, A280938.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1-x^(7*(2*k-1))) * (1-x^(7*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 2*Pi * BesselI(1, Pi/6 * sqrt(11*(24*n-1)/14)) / (7*sqrt((24*n-1)/11)).

a(n) ~ exp(Pi * sqrt(11*n/21)) * 11^(1/4) / (2 * 3^(1/4) * 7^(3/4) * n^(3/4)) * (1 -(3*sqrt(21)/(8*Pi*sqrt(11)) + Pi*sqrt(11)/(48*sqrt(21)))/sqrt(n) + (11*Pi^2/96768 - 315/(1408*Pi^2) + 5/128)/n).