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
Keywords
Examples
1 - 9*q + 54*q^2 - 324*q^3 + 1989*q^4 - 12204*q^5 + 74844*q^6 - 459072*q^7 + ...
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.
Programs
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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 *)
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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))}
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).
a(n) ~ (-1)^n * 8 * sqrt(3) * Pi^(5/2) * exp(Pi*n/sqrt(3)) / Gamma(1/6)^3. - Vaclav Kotesovec, Nov 14 2015
Comments