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%I #24 May 13 2021 12:16:31
%S 1,-3,0,5,0,0,-7,0,0,3,0,0,15,0,0,-32,0,0,9,0,0,58,0,0,-96,0,0,22,0,0,
%T 149,0,0,-253,0,0,68,0,0,372,0,0,-599,0,0,140,0,0,826,0,0,-1317,0,0,
%U 317,0,0,1768,0,0,-2735,0,0,632,0,0,3526,0,0,-5434,0,0
%N Expansion of (eta(q) / eta(q^9))^3 in powers of q.
%C Number 4 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Jul 21 2014
%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(9). [Yang 2004] - _Michael Somos_, Jul 21 2014
%H Seiichi Manyama, <a href="/A131986/b131986.txt">Table of n, a(n) for n = -1..10000</a>
%H Y. Yang, <a href="http://dx.doi.org/10.1112/S0024609304003510">Transformation formulas for generalized Dedekind eta functions</a>, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
%F Euler transform of period 9 sequence [ -3, -3, -3, -3, -3, -3, -3, -3, 0, ...].
%F G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u + v)^3 - u*v * (27 + 9*(u+v) + u*v).
%F G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = u^2 + w^2 + u*w - v^2*(u+w) - 6*v^2 - 6*v*(u+w) - 27*v.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 27 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121589.
%F a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A058091(n).
%F G.f.: (1/x) * (Product_{k>0} (1 - x^k) / (1 - x^(9*k)))^3.
%F Convolution inverse of A121589. - _Michael Somos_, Jul 21 2014
%F Convolution cube of A062246. - _Michael Somos_, Nov 03 2015
%F a(-1) = 1, a(n) = -(3/(n+1))*Sum_{k=1..n+1} A116607(k)*a(n-k) for n > -1. - _Seiichi Manyama_, Mar 29 2017
%F G.f. A(q) satisfies 0 = f(A(q), A(q^3)) where f(u, v) = (27 + 9*u + u^2) * (27 + 9*v + v^2) * u - v^3. - _Michael Somos_, May 13 2021
%e G.f. = 1/q - 3 + 5*q^2 - 7*q^5 + 3*q^8 + 15*q^11 - 32*q^14 + 9*q^17 + ...
%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^9]))^3, {q, 0, n}]; (* _Michael Somos_, Jul 21 2014 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^9 + A))^3, n))};
%Y Cf. A062246, A058091, A121589.
%K sign
%O -1,2
%A _Michael Somos_, Aug 04 2007
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