A131986 Expansion of (eta(q) / eta(q^9))^3 in powers of q.

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, 149, 0, 0, -253, 0, 0, 68, 0, 0, 372, 0, 0, -599, 0, 0, 140, 0, 0, 826, 0, 0, -1317, 0, 0, 317, 0, 0, 1768, 0, 0, -2735, 0, 0, 632, 0, 0, 3526, 0, 0, -5434, 0, 0

Offset

-1,2

Comments

Number 4 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014

A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(9). [Yang 2004] - Michael Somos, Jul 21 2014

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

Formula

Euler transform of period 9 sequence [ -3, -3, -3, -3, -3, -3, -3, -3, 0, ...].

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).

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.

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.

a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A058091(n).

G.f.: (1/x) * (Product_{k>0} (1 - x^k) / (1 - x^(9*k)))^3.

Convolution inverse of A121589. - Michael Somos, Jul 21 2014

Convolution cube of A062246. - Michael Somos, Nov 03 2015

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

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

Example

G.f. = 1/q - 3 + 5*q^2 - 7*q^5 + 3*q^8 + 15*q^11 - 32*q^14 + 9*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^9]))^3, {q, 0, n}]; (* Michael Somos, Jul 21 2014 *)

Prog

(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))};

Crossrefs

Cf. A062246, A058091, A121589.

Sequence in context: A198954 A136599 A227498 * A002656 A234434 A234020

Adjacent sequences: A131983 A131984 A131985 * A131987 A131988 A131989

Keyword

sign

Author

Michael Somos, Aug 04 2007

Status

approved