A341714 - OEIS

%I #18 Aug 01 2022 06:49:23

%S 1,1,2,3,5,7,11,15,22,30,42,56,77,100,134,174,228,292,378,479,612,770,

%T 972,1213,1519,1881,2334,2874,3540,4331,5302,6450,7848,9501,11496,

%U 13851,16680,20006,23980,28648,34193,40689,48378,57360,67948,80295,94788,111652,131388,154293

%N Coefficients in the expansion of Product_{m>=1} (1 - q^(13*m))/(1 - q^m).

%H Seiichi Manyama, <a href="/A341714/b341714.txt">Table of n, a(n) for n = 0..10000</a>

%H J. J. Webb, <a href="https://doi.org/10.1007/s11139-010-9227-4">Arithmetic of the 13-regular partition function modulo 3</a>, Ramanujan Journal, 25 (2011), 49-56.

%F a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), set s=13. - _Vaclav Kotesovec_, Aug 01 2022

%Y Column k=13 of A286653.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Feb 21 2021