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).
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, 33, 36, 42, 46, 53, 58, 67, 73, 83, 91, 104, 113, 128, 140, 158, 173, 194, 212, 238, 260, 290, 317, 353, 385, 428, 467, 517, 564
Offset: 0
Keywords
Examples
From _Wolfdieter Lang_, Nov 02 2016: (Start) a(0) = 1 from the n=0 sum term (undefined product put to 1), a(n) = 1 for n=4..9 from the partition [n-2,2], a(10) = 2 from [8,2] and [4,4,2], a(11) = 2 from [9,2] and [5,4,2], a(12) = 3 from [10,2], [6,4,2], [5,5,2], 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]. The maximal number of parts needed for n is floor((-1+sqrt(9+4*n))/2) = A259361(n+2). A simpler interpretation: 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]. The maximal number of parts needed for n is floor((-3+sqrt(9+4*n))/2). (End)
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
- George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
- Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
- Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
- James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, The Ramanujan Journal 29.1-3 (2012): 199-211.
Crossrefs
Programs
-
Mathematica
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 *)
Formula
From Wolfdieter Lang, Nov 02 2016: (Start)
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).
G.f.: Sum_{m >= 0} q^(m*(m+3)) / Product_{j=1..m} (1-q^j) from (AB) eq. 51.
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)
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
Comments