A035954 Number of partitions of n into parts not of the form 13k, 13k+6 or 13k-6. Also number of partitions with at most 5 parts of size 1 and differences between parts at distance 5 are greater than 1.

1, 1, 2, 3, 5, 7, 10, 13, 19, 25, 34, 44, 59, 75, 98, 124, 159, 199, 252, 312, 391, 481, 595, 727, 893, 1084, 1320, 1594, 1928, 2315, 2784, 3325, 3977, 4730, 5627, 6664, 7894, 9310, 10981, 12905, 15162, 17756, 20787, 24263, 28310, 32946, 38317, 44462

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

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

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Programs

Mathematica

nmax = 60; CoefficientList[Series[Product[1 / ((1 - x^(13*k-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-8)) * (1 - x^(13*k-9)) * (1 - x^(13*k-10)) * (1 - x^(13*k-11)) * (1 - x^(13*k-12)) ), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 22 2015 *)

Formula

a(n) ~ sin(6*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

Extensions

a(0)=1 prepended by Seiichi Manyama, May 08 2018