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A264594 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[7](q).

%I A264594 #13 Dec 24 2016 10:31:58
%S A264594 1,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,8,9,10,12,13,
%T A264594 15,17,19,21,24,26,29,32,36,39,44,48,54,59,66,72,81,88,98,107,119,129,
%U A264594 143,156,172,187,206,224,247,268,294,320,351,381,417,453,495,537,586,635,693,750,816
%N A264594 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[7](q).
%C A264594 It is conjectured that G[i](q) = 1 + O(q^i) for all i.
%H A264594 Vaclav Kotesovec, <a href="/A264594/b264594.txt">Table of n, a(n) for n = 0..1000</a>
%H A264594 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 A264594 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 A264594 a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(11/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Dec 24 2016
%t A264594 nmax = 100; CoefficientList[Series[Sum[x^(k*(k+6))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 24 2016 *)
%Y A264594 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 A264594 nonn
%O A264594 0,17
%A A264594 _N. J. A. Sloane_, Nov 19 2015