A104410 Coefficients of the C-Rogers-Selberg identity.

1, -1, 1, -1, 2, -2, 2, -3, 4, -4, 5, -6, 8, -9, 10, -12, 15, -17, 19, -22, 27, -30, 34, -39, 46, -52, 58, -66, 77, -86, 96, -109, 125, -139, 155, -174, 198, -220, 244, -273, 308, -341, 377, -420, 470, -519, 573, -635, 707, -779, 857, -946, 1049, -1152, 1264, -1392, 1536, -1683, 1843, -2022, 2224

Offset

0,5

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, -q^6) / f(-q^2) in powers of q where f() is Ramanujan's theta function.

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

a(n) ~ (-1)^n * cos(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 + q^2 - q^3 + 2*q^4 - 2*q^5 + 2*q^6 - 3*q^7 + 4*q^8 - 4*q^9 + 5*q^10 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1-x^(7*k-1))*(1-x^(7*k-6))*(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, 1, -1, 0, -1, 0, 0, 1, 0, 0, -1, 0, -1, 1][k%14+1]), n))} /* Michael Somos, Dec 04 2007 */

Crossrefs

Cf. A104408, A104409.

Sequence in context: A076269 A143644 A363336 * A018048 A077564 A088044

Adjacent sequences: A104407 A104408 A104409 * A104411 A104412 A104413

Keyword

sign,changed

Author

Eric W. Weisstein, Mar 06 2005

Status

approved