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).

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, 18, 21, 23, 26, 28, 33, 35, 40, 44, 50, 54, 62, 67, 76, 83, 93, 101, 114, 123, 138, 150, 167, 181, 202, 219, 243, 264, 292, 317, 351, 380, 419, 455, 500, 542, 596, 645, 707, 766, 838, 907, 992, 1072

Offset

0,15

Comments

It is conjectured that G[i](q) = 1 + O(q^i) for all i.

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

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 10 2016: (Start)

G.f.: G[6](q) = GII_2(q) = Sum_{m>=0} q^(m*(m+5)) / Product_{j =1..m} (1 - q^j).

See Andrews and Baxter [A-B], eq. (5.1) for i=6.

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)

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

Example

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

Mathematica

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 *)

Crossrefs

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.

Sequence in context: A025154 A241827 A096401 * A026828 A025153 A026803

Adjacent sequences: A264590 A264591 A264592 * A264594 A264595 A264596

Keyword

nonn

Author

N. J. A. Sloane, Nov 19 2015

Status

approved