OFFSET
0,11
COMMENTS
It is conjectured that G[i](q) = 1 + O(q^i) for all i.
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
From Wolfdieter Lang, Nov 02 2016: (Start)
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.
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.
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.
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)
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.
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
EXAMPLE
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)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 18 2015
STATUS
approved