A264595 - OEIS

A264595

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[8](q).

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 9, 9, 11, 12, 14, 15, 18, 19, 22, 24, 27, 29, 33, 35, 40, 43, 48, 52, 59, 63, 71, 77, 86, 93, 104, 112, 125, 135, 149, 161, 179, 192, 212, 229, 252, 272, 299, 322, 354, 382, 418, 451, 494, 532, 581, 627, 683

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OFFSET

0,19

COMMENTS

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

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

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

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * sqrt(5) * phi^(13/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 24 2016

MATHEMATICA

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+7))/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[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595.

Sequence in context: A025160 A026831 A096740 * A026830 A025155 A286744

Adjacent sequences: A264592 A264593 A264594 * A264596 A264597 A264598

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 19 2015

STATUS

approved