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Number of partitions of n into parts not of the form 19k, 19k+8 or 19k-8. Also number of partitions with at most 7 parts of size 1 and differences between parts at distance 8 are greater than 1.
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%I #8 May 10 2018 03:15:38

%S 1,2,3,5,7,11,15,21,29,40,52,71,92,121,156,202,256,328,412,520,649,

%T 811,1002,1243,1526,1875,2289,2794,3388,4112,4960,5982,7183,8619,

%U 10299,12302,14638,17404,20630,24431,28848,34038,40053,47088,55234

%N Number of partitions of n into parts not of the form 19k, 19k+8 or 19k-8. Also number of partitions with at most 7 parts of size 1 and differences between parts at distance 8 are greater than 1.

%C Case k=9,i=8 of Gordon Theorem.

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

%F a(n) ~ exp(4*Pi*sqrt(2*n/57)) * 2^(3/4) * cos(3*Pi/38) / (3^(1/4) * 19^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, May 10 2018

%t nmax = 60; Rest[CoefficientList[Series[Product[(1 - x^(19*k))*(1 - x^(19*k+ 8-19))*(1 - x^(19*k- 8))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, May 10 2018 *)

%K nonn,easy

%O 1,2

%A _Olivier GĂ©rard_