A035953 Number of partitions of n into parts not of the form 13k, 13k+5 or 13k-5. Also number of partitions with at most 4 parts of size 1 and differences between parts at distance 5 are greater than 1.
1, 2, 3, 5, 6, 10, 13, 18, 24, 33, 42, 57, 72, 94, 119, 153, 190, 242, 299, 374, 460, 570, 695, 855, 1036, 1262, 1523, 1843, 2210, 2660, 3175, 3797, 4514, 5372, 6357, 7533, 8880, 10474, 12306, 14459, 16925, 19818, 23125, 26981, 31392, 36512, 42355
Offset: 1
References
- G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
Programs
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Mathematica
nmax = 60; Rest[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-6)) * (1 - x^(13*k-7)) * (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(5*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
Comments