A035296 - OEIS

%I #21 Jul 08 2025 21:31:58

%S 1,1,2,4,7,12,18,28,41,60,85,119,164,225,304,408,542,716,938,1222,

%T 1582,2037,2609,3326,4220,5332,6708,8407,10497,13061,16197,20020,

%U 24671,30313,37141,45383,55311,67242,81552,98678,119135,143522,172545,207018,247899,296294,353492

%N Expansion of sum ( q^n / product( 1-q^k, k=1..4*n), n=0..inf ).

%H Vaclav Kotesovec, <a href="/A035296/b035296.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ Gamma(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(29/8) * 3^(1/8) * Pi^(3/4) * n^(5/8)). - _Vaclav Kotesovec_, Jun 17 2025

%t nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* _Vaclav Kotesovec_, Jun 16 2025 *)

%t nmax = 50; p=1; s=1; Do[p=Expand[p*(1-x^(4*k))*(1-x^(4*k-1))*(1-x^(4*k-2))*(1-x^(4*k-3))];p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]];s+=x^k/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 16 2025 *)

%Y Cf. A363067, A385068.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Vaclav Kotesovec_, Jun 16 2025