cp's OEIS Frontend

A109389 Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q.

%I A109389 #32 Mar 26 2017 07:23:14
%S A109389 1,-1,0,0,0,-1,1,-1,1,0,0,-1,2,-2,1,0,1,-2,3,-3,2,-1,1,-3,5,-5,3,-1,2,
%T A109389 -5,7,-7,5,-3,3,-7,11,-11,7,-4,6,-11,15,-15,11,-7,8,-15,22,-22,15,-10,
%U A109389 13,-22,30,-30,23,-16,18,-30,42,-42,31,-22,27,-43,56,-56,44,-33,37,-57,77,-77,59,-45,53,-79,101,-101,82,-64
%N A109389 Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q.
%C A109389 In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - _Vaclav Kotesovec_, Aug 31 2015
%H A109389 Seiichi Manyama, <a href="/A109389/b109389.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Vaclav Kotesovec)
%H A109389 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], Sep 30 2015, p. 14.
%F A109389 Euler transform of period 6 sequence [ -1, 0, 0, 0, -1, 0, ...].
%F A109389 G.f.: 1/(Product_{k>0} (1+x^(2k-1)+x^(4k-2))) = Product_{k>0} (1-x^(6k-1))(1-x^(6k-5)) = Product_{k>0} (1-x^k+x^(2k)) (where 1-x+x^2 is 6th cyclotomic polynomial).
%F A109389 Given g.f. A(x), then B(x)=x*A(x^12) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=(v^2+u^4)*(v^2+w^4)-2*v^4*(1-v*u^2*w^2).
%F A109389 Expansion of G(x^6) * H(x) - x * G(x) * H(x^6) where G(), H() are Rogers-Ramanujan functions.
%F A109389 a(n) = (-1)^n*A098884(n).
%F A109389 a(n) ~ (-1)^n * exp(sqrt(n)*Pi/3) / (2*sqrt(6)*n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015
%F A109389 a(n) = -(1/n)*Sum_{k=1..n} A186099(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 26 2017
%e A109389 q - q^13 - q^61 + q^73 - q^85 + q^97 - q^133 + 2*q^145 - 2*q^157 + q^169 + ...
%t A109389 nmax = 100; CoefficientList[Series[Product[(1 + x^(3*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 30 2015 *)
%t A109389 QP = QPochhammer; s = QP[q]*(QP[q^6]/(QP[q^2]*QP[q^3])) + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 23 2015 *)
%o A109389 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)/eta(x^2+A)/eta(x^3+A), n))}
%Y A109389 Cf. A098884.
%Y A109389 Cf. A081360 (m=2), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).
%K A109389 sign
%O A109389 0,13
%A A109389 _Michael Somos_, Jun 26 2005