A280908 Expansion of Product_{k>=1} ((1+x^k) / ((1-x^(2*k-1)) * (1-x^(8*k-4)))).

1, 2, 3, 6, 10, 16, 25, 38, 56, 82, 118, 166, 233, 322, 440, 598, 804, 1072, 1422, 1872, 2449, 3188, 4126, 5312, 6810, 8690, 11040, 13974, 17618, 22130, 27707, 34572, 43000, 53328, 65942, 81312, 100004, 122674, 150110, 183254, 223200, 271248, 328945, 398086

Offset

0,2

References

G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook Part II, Springer, 2009. [Entry 1.7.5]

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.

Formula

a(n) ~ sqrt(3)*Pi * BesselI(1, sqrt(12*n+3)*Pi/4) / (4*sqrt(8*n+2)).

a(n) ~ 3^(1/4) * exp(sqrt(3*n)*Pi/2) / (8*sqrt(2)*n^(3/4)) * (1 + (Pi/16 - 1/(4*Pi))*sqrt(3/n) + (3*Pi^2/512 - 5/(32*Pi^2) - 15/64)/n).

Mathematica

nmax = 50; CoefficientList[Series[Product[(1+x^k) / ((1-x^(2*k-1)) * (1-x^(8*k-4))), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Sequence in context: A024801 A324742 A260599 * A146163 A101277 A262984

Adjacent sequences: A280905 A280906 A280907 * A280909 A280910 A280911

Keyword

nonn

Author

Vaclav Kotesovec, Jan 10 2017

Status

approved