A104409 Coefficients of the B-Rogers-Selberg identity.

1, 0, 0, 0, 1, -1, 1, -1, 2, -2, 2, -2, 4, -4, 4, -5, 7, -7, 8, -9, 12, -13, 14, -16, 21, -22, 24, -28, 34, -37, 41, -46, 55, -60, 66, -74, 87, -95, 104, -117, 135, -147, 162, -180, 205, -225, 246, -273, 309, -337, 369, -408, 457, -499, 546, -601, 669, -730, 796, -874, 969, -1055, 1149, -1259

Offset

0,9

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

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^2, -q^5) / f(-q^2) in powers of q where f() is Ramanujan's theta function.

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

a(n) ~ (-1)^n * sin(Pi/7) * 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^4 - q^5 + q^6 - q^7 + 2*q^8 - 2*q^9 + 2*q^10 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1-x^(7*k-2))*(1-x^(7*k-5))*(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, 0, 0, -1, 1, -1, 1, -1, 1, -1, 0, 0, 0][k%14+1]), n))} /* Michael Somos, Dec 04 2007 */

Crossrefs

Cf. A104408, A104410.

Sequence in context: A200675 A029079 A035398 * A274144 A214628 A355806

Adjacent sequences: A104406 A104407 A104408 * A104410 A104411 A104412

Keyword

sign

Author

Eric W. Weisstein, Mar 06 2005

Status

approved