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A264595 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[8](q).

%I A264595 #14 Dec 24 2016 10:33:13
%S A264595 1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,9,9,11,
%T A264595 12,14,15,18,19,22,24,27,29,33,35,40,43,48,52,59,63,71,77,86,93,104,
%U A264595 112,125,135,149,161,179,192,212,229,252,272,299,322,354,382,418,451,494,532,581,627,683
%N A264595 Let G[1](q) denote the g.f. for A003114 and G[2](q) the g.f. for A003106 (the two Rogers-Ramanujan identities). For i>=3, let G[i](q) = (G[i-1](q)-G[i-2](q))/q^(i-2). Sequence gives coefficients of G[8](q).
%C A264595 It is conjectured that G[i](q) = 1 + O(q^i) for all i.
%H A264595 Vaclav Kotesovec, <a href="/A264595/b264595.txt">Table of n, a(n) for n = 0..1000</a>
%H A264595 Shashank Kanade, <a href="http://www.math.rutgers.edu/~skanade/SK-Defense-Handout.pdf">Some results on the representation theory of vertex operator algebras and integer partition identities</a>, PhD Handout, Math. Dept., Rutgers University, April 2015.
%H A264595 Shashank Kanade, <a href="http://dx.doi.org/doi:10.7282/T3TX3H7B">Some results on the representation theory of vertex operator algebras and integer partition identities</a>, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
%F A264595 a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(13/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Dec 24 2016
%t A264595 nmax = 100; CoefficientList[Series[Sum[x^(k*(k+7))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 24 2016 *)
%Y A264595 For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595.
%K A264595 nonn
%O A264595 0,19
%A A264595 _N. J. A. Sloane_, Nov 19 2015