OFFSET
0,13
COMMENTS
It is conjectured that G[i](q) = 1 + O(q^i) for all i.
From Wolfdieter Lang, Nov 03 2016: (Start)
The generalized Rogers-Ramanujan [R-R] series G[i](q) of Andrews and Baxter [A-B] have a standard combinatorial interpretation of the Schur and MacMahon type (see Hardy [H] and Hardy-Wright [H-W] for the original [R-R] case) inferred from the formula G[i](q) = Sum_{m >=0} q^(m*(m+i-1)}/Product_{j =1..m} (1 - q^j) ([A-B], eq. (5.1)). Define GI_k(q) = G[2*k+1](q) and GII_k(q) = G[2*k](q), for k = 0, 1,..., and prove the two formulas I(m,k): m*(m+2*k) = Sum_{j = 1..2*m-1} (2*k + j), and II(m,k): m*(m+2*k+1) = Sum_{j = 1..m} (2*(k + j)) for fixed positive m by induction over k = 0, 1, ... . For GI_k(q) define the special m-part partition SPI(m,k) = [2*k+2*m-1,2*k+2*m-3,...,2*k+1] of m*(m+2*k), and for GII_k(q) the special m-part partition SPII(m,k) [2*(k+1),2*(k+2),...,2*(k+1))] of m*(m+2*k+1).
Then GI_k(q) = 1 + Sum_{n >=1} aI(k,n)*q^n with aI(k,n) the number of partitions of n without parts 1, 2, ..., 2*k, and the parts differ by at least 2. GII_k(q) = 1 + Sum_{n >=1} aII(k,n)*q^n with aII(k,n) the number of partitions of n without parts 1, 2, ..., 2*k+1, and the parts differ by at least 2. The proof can be directly adapted from the one given in [H] or [H-W] for k=1.
For the partitions of n generated by GI_k(q) one needs the maximal part number MmaxI(k,n) = floor(-k + sqrt(k^2 + n)). For the GII_k(q) case MmaxII(k,n) = floor(-(2*k+1) + sqrt((2*k+1)^2 + 4*n)).
The present sequence is aI(2,n), in [A-B] notation generated by G[5](q), giving the number of partitions of n without parts 1, 2, 3 and 4, and the parts differ by at least 2.
(End)
REFERENCES
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 91-92.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 290-291.
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.
FORMULA
From Wolfdieter Lang, Nov 03 2016: (Start)
G.f.: G[5](q) = GI_2(q) = Sum_{m >=0} q^(m*(m+4)) / Product_{j =1..m} (1 - q^j).
See [A-B], eq. (5.1) for i=5.
a(0) = 1 and a(n) gives the number of partitions of n without part 1 and 2, the parts differing by at least 2.
G.f.: Sum_{m=0} ((-1)^m*(1 - q^(m+1))*(1 - q^(m+2))*(1 - q^(m+3))*(1 - q^(2*(m+2))) * q^(5*(n+3)*n/2)) / Product_{j>=1} (1 - q^j). See [A-B], eq. (3.8) for i=5. (End)
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(7/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016
EXAMPLE
From Wolfdieter Lang, Nov 03 2016: (Start)
a(5) = 1 because the only partition of n = 5 without parts 1, 2, 3 and 4, and parts differing by at least 2 is [5].
a(12) = 2 from the two partitions [12] and [7,5] of n = 12.
a(18) = 5 from the five partitions [18], [13,5], [12,6], [11,7], [10,8] of n = 18.
(End)
MATHEMATICA
nmax = 100; CoefficientList[Series[Sum[x^(k*(k+4))/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 19 2015
STATUS
approved