A104408 Coefficients of the A-Rogers-Selberg identity.

1, 0, 1, -1, 1, -1, 2, -2, 3, -3, 4, -5, 6, -6, 8, -9, 11, -12, 15, -17, 20, -22, 26, -30, 35, -38, 45, -51, 58, -64, 74, -83, 95, -105, 119, -134, 151, -166, 188, -210, 235, -259, 291, -323, 360, -396, 441, -489, 543, -595, 661, -730, 805, -883, 976, -1073, 1182, -1293, 1423, -1562, 1714

Offset

0,7

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

J. Mc Laughlin, A. V. Sills and P. Zimmer, Rogers-Ramanujan-Slater Type Identities, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008. See "2.7 The Rogers-Selberg Mod 7 Identities".

Eric Weisstein's World of Mathematics, Rogers-Selberg Identities

Formula

Expansion of f(-q^3, -q^4) / f(-q^2) in powers of q where f() is Ramanujan's theta function.

Euler transform of period 14 sequence [ 0, 1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, ...]. - Michael Somos, Dec 04 2007

a(n) ~ (-1)^n * cos(3*Pi/14) * 11^(1/4) * exp(Pi*sqrt(11*n/42)) / (3^(1/4) * 14^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 04 2015

Example

1 + q^2 - q^3 + q^4 - q^5 + 2*q^6 - 2*q^7 + 3*q^8 - 3*q^9 + 4*q^10 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1-x^(7*k-3))*(1-x^(7*k-4))*(1-x^(7*k))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[0, 0, -1, 1, 0, 0, -1, 1, -1, 0, 0, 1, -1, 0][k%14+1]), n))} /* Michael Somos, Dec 04 2007 */

Crossrefs

Cf. A104409, A104410.

Sequence in context: A337766 A029071 A117144 * A008718 A248958 A030719

Adjacent sequences: A104405 A104406 A104407 * A104409 A104410 A104411

Keyword

sign

Author

Eric W. Weisstein, Mar 05 2005

Status

approved