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A264593 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[6](q).

%I A264593 #25 Dec 24 2016 10:30:48
%S A264593 1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,6,8,8,10,11,13,14,17,
%T A264593 18,21,23,26,28,33,35,40,44,50,54,62,67,76,83,93,101,114,123,138,150,
%U A264593 167,181,202,219,243,264,292,317,351,380,419,455,500,542,596,645,707,766,838,907,992,1072
%N A264593 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[6](q).
%C A264593 It is conjectured that G[i](q) = 1 + O(q^i) for all i.
%C A264593 For n >=1 a(n) gives the number of partitions of n without parts 1, 2, 3, 4, and 5, and the parts differ by at least 2. For the proof see a comment given in A264592. - _Wolfdieter Lang_, Nov 10 2016
%H A264593 Vaclav Kotesovec, <a href="/A264593/b264593.txt">Table of n, a(n) for n = 0..1000</a>
%H A264593 George E. Andrews; R. J. Baxter, <a href="http://www.computing-wisdom.com/jstor/rogers-ramanujan.pdf">A motivated proof of the Rogers-Ramanujan identities</a>, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
%H A264593 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 A264593 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 A264593 From _Wolfdieter Lang_, Nov 10 2016: (Start)
%F A264593 G.f.: G[6](q) = GII_2(q) = Sum_{m>=0} q^(m*(m+5)) / Product_{j =1..m} (1 - q^j).
%F A264593 See Andrews and Baxter [A-B], eq. (5.1) for i=6.
%F A264593 G.f.: Sum_{m=0} ((-1)^m*(1 - q^(m+1))*(1 - q^(m+2))*(1 - q^(m+3))*(1 - q^(m+4))*(1 - q^(2*m+5))*q^((5*m+19)*m/2)) / Product_{j>=1} (1 - q^j). See [A-B] eq. (3.8) for i=6. (End)
%F A264593 a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(9/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Dec 24 2016
%e A264593 a(18) = 4 because the four partitions of 18 without parts 1, 2, 3, 4 and 5, and the parts differ by at least 2 are [18], [12, 6], [11, 7], [10, 8]. - _Wolfdieter Lang_, Nov 10 2016
%t A264593 nmax = 100; CoefficientList[Series[Sum[x^(k*(k+5))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 24 2016 *)
%Y A264593 For the generalized Rogers-Ramanujan series G[0], G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003113, A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595.
%K A264593 nonn
%O A264593 0,15
%A A264593 _N. J. A. Sloane_, Nov 19 2015