A115784 Expansion of b(q) / a(q) in powers of q of cubic AGM theta function.

1, -9, 54, -324, 1989, -12204, 74844, -459072, 2815830, -17271468, 105938118, -649793448, 3985642908, -24446767374, 149949318096, -919745243064, 5641448209173, -34602992662356, 212244632371188, -1301846473509156, 7985145356345268, -48978545212087776

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), 35-47.

Formula

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - u*v)^3 - (1 - u^3) * (1 - v^3).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 + 2*u)^3 * v^3 - 9 * u * (1 + u + u^2).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (1 + 2*u1) * (1 + 2*u2) * u3*u6 - 3 * (u1 + u2 + u1*u2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058091.

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

Convolution inverse is A215690. Convolution with A004016 is A005928.

a(n) ~ (-1)^n * 8 * sqrt(3) * Pi^(5/2) * exp(Pi*n/sqrt(3)) / Gamma(1/6)^3. - Vaclav Kotesovec, Nov 14 2015

Example

1 - 9*q + 54*q^2 - 324*q^3 + 1989*q^4 - 12204*q^5 + 74844*q^6 - 459072*q^7 + ...

Mathematica

QP = QPochhammer;  s = QP[q]^3/(QP[q]^3 + 9*q*QP[q^9]^3) + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^3/ (eta[q]^3 + 9*eta[q^9]^3), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 11 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3), n))}

Crossrefs

Cf. A004016, A005928, A215690, A258942.

Sequence in context: A276602 A079764 A079761 * A037599 A037704 A093847

Adjacent sequences: A115781 A115782 A115783 * A115785 A115786 A115787

Keyword

sign

Author

Michael Somos, Jan 31 2006

Status

approved