A035949 Number of partitions in parts not of the form 13k, 13k+1 or 13k-1. Also number of partitions with no part of size 1 and differences between parts at distance 5 are greater than 1.

0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 23, 32, 39, 51, 61, 80, 95, 122, 146, 183, 219, 273, 324, 399, 475, 578, 685, 830, 979, 1177, 1387, 1655, 1945, 2311, 2705, 3198, 3737, 4396, 5121, 6003, 6973, 8143, 9439, 10981, 12697, 14730, 16987, 19648, 22614

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Case k=6,i=1 of Gordon Theorem.

References

• G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Programs

Mathematica

nmax = 60; Rest[CoefficientList[Series[Product[1 / ((1 - x^(13*k-2)) * (1 - x^(13*k-3)) * (1 - x^(13*k-4)) * (1 - x^(13*k-5)) * (1 - x^(13*k-6)) * (1 - x^(13*k-7)) * (1 - x^(13*k-8)) * (1 - x^(13*k-9)) * (1 - x^(13*k-10)) * (1 - x^(13*k-11)) ), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 22 2015 *)

Formula

a(n) ~ sin(Pi/13) * 5^(1/4) * exp(2*Pi*sqrt(5*n/39)) / (3^(1/4) * 13^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 22 2015