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Coefficients of the A-Rogers-Selberg identity.
3

%I #26 Oct 20 2019 03:11:25

%S 1,0,1,-1,1,-1,2,-2,3,-3,4,-5,6,-6,8,-9,11,-12,15,-17,20,-22,26,-30,

%T 35,-38,45,-51,58,-64,74,-83,95,-105,119,-134,151,-166,188,-210,235,

%U -259,291,-323,360,-396,441,-489,543,-595,661,-730,805,-883,976,-1073,1182,-1293,1423,-1562,1714

%N Coefficients of the A-Rogers-Selberg identity.

%H Seiichi Manyama, <a href="/A104408/b104408.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.

%H J. Mc Laughlin, A. V. Sills and P. Zimmer, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS15">Rogers-Ramanujan-Slater Type Identities</a>, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008. See "2.7 The Rogers-Selberg Mod 7 Identities".

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-SelbergIdentities.html">Rogers-Selberg Identities</a>

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

%F 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

%F 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

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

%t 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 *)

%o (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 */

%Y Cf. A104409, A104410.

%K sign

%O 0,7

%A _Eric W. Weisstein_, Mar 05 2005