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A264591 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[4](q).

%I A264591 #42 Mar 05 2025 02:39:19
%S A264591 1,0,0,0,1,1,1,1,1,1,2,2,3,3,4,4,5,5,7,7,9,10,12,13,16,17,20,22,26,28,
%T A264591 33,36,42,46,53,58,67,73,83,91,104,113,128,140,158,173,194,212,238,
%U A264591 260,290,317,353,385,428,467,517,564
%N A264591 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[4](q).
%C A264591 It is conjectured that G[i](q) = 1 + O(q^i) for all i.
%C A264591 For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[4](x). - _N. J. A. Sloane_, Nov 22 2015
%C A264591 From _Wolfdieter Lang_, Nov 02 2016: (Start)
%C A264591 The second g.f. given below leads to a combinatorial partition interpretation from (2 + 4 + ... + 2*m) + 2*m = m*(m+3). Take for the sum term m the special M=m+1 part partition [2m,2m,2*(m-1),...,4,2] together with arbitrary partitions of N with part number m' <= M-1 = m added to the first m' parts.
%C A264591 Summing over m>=1 leads to partitions of n = m*(m+3) + N which have no part 1, only one part 2 except for n=4 and for number of parts M >= 3 the difference of parts except of the first two parts has to be at least 2. See the examples below.
%C A264591 A simpler interpretation uses m*(m+3) = 4 + 6 + ... + 2*(m+1), leading to a(n) as the number of partitions of n with parts >= 4 and parts differing by at least 2.
%C A264591 This is in the spirit of MacMahon's and Schur's interpretation of the sum version of the Rogers-Ramanujan identities. See the Hardy and Hardy-Wright references under A003114. (End)
%H A264591 Vaclav Kotesovec, <a href="/A264591/b264591.txt">Table of n, a(n) for n = 0..1000</a>
%H A264591 George E. Andrews; R. J. Baxter, <a href="https://web.archive.org/web/20161027143824/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 A264591 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 A264591 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.
%H A264591 James Lepowsky and Minxian Zhu, <a href="http://arxiv.org/abs/1205.6570">A motivated proof of Gordon's identities</a>, The Ramanujan Journal 29.1-3 (2012): 199-211.
%F A264591 From _Wolfdieter Lang_, Nov 02 2016: (Start)
%F A264591 G.f.: G[4](q) = (Sum_{n >= 0} (-1)^n*(1 - q^(n+1))*(1 - q^(n+2))*(1 - q^(2*n+3))*q^((5*n+11)*n/2)) / (Product_{j >= 1} (1 - q^j)), from the Andrews-Baxter (AB) reference, eq. (3.7).
%F A264591 G.f.: Sum_{m >= 0} q^(m*(m+3)) / Product_{j=1..m} (1-q^j) from (AB) eq. 51.
%F A264591 This can also be derived from the Hardy (H) or Hardy-Wright reference (see A006141): Put G_4(a,q):= (H_1(a,q) - H_1(a*q,q)) / (a*q) with H_1(a,x) from (H) p. 95, first eq. Then G[4](q) = G_4(q,q). (End)
%F A264591 a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(5/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Dec 24 2016
%e A264591 From _Wolfdieter Lang_, Nov 02 2016: (Start)
%e A264591 a(0) = 1 from the n=0 sum term (undefined product put to 1),
%e A264591 a(n) = 1 for n=4..9 from the partition [n-2,2],
%e A264591 a(10) = 2 from [8,2] and [4,4,2],
%e A264591 a(11) = 2 from [9,2] and [5,4,2],
%e A264591 a(12) = 3 from [10,2], [6,4,2], [5,5,2],
%e A264591 a(18) = 7 from [16,2], all 1+4=5 partitions of 18-10 = 8 with part number <= 2 added to the first two part of [4,4,2] and the new four part partition [6,6,4,2].
%e A264591 The maximal number of parts needed for n is floor((-1+sqrt(9+4*n))/2) = A259361(n+2).
%e A264591 A simpler interpretation:
%e A264591 a(18) = 7 from the partitions of 18 with parts >=4 and parts differing by at least 2: [18], [14,4], [13,5], [12,6], [11,7], [10,8], [8,6,4].
%e A264591 The maximal number of parts needed for n is floor((-3+sqrt(9+4*n))/2).
%e A264591 (End)
%t A264591 nmax = 100; CoefficientList[Series[Sum[x^(k*(k+3))/Product[1-x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 24 2016 *)
%Y A264591 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. G[0] = G[1]+G[2] is given by A003113.
%K A264591 nonn
%O A264591 0,11
%A A264591 _N. J. A. Sloane_, Nov 18 2015