A215690 Expansion of a(q) / b(q) in powers of q where a(), b() are cubic AGM theta functions.
1, 9, 27, 81, 198, 459, 972, 1989, 3861, 7290, 13284, 23679, 41148, 70218, 117504, 193671, 314262, 503415, 796068, 1244988, 1925910, 2950668, 4478328, 6739497, 10059228, 14901471, 21914442, 32011119, 46456272, 67010679, 96093864, 137039922, 194395221
Offset: 0
Keywords
Examples
1 + 9*q + 27*q^2 + 81*q^3 + 198*q^4 + 459*q^5 + 972*q^6 + 1989*q^7 + 3861*q^8 + ...
References
- O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See u, page 1.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- 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 = 1 + 9*q*(QP[q^9]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
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PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 + 9 * x * (eta(x^9 + A) / eta(x + A))^3, n))}
Formula
Expansion of 1 + 3 * c(q^3) / b(q) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of 1 + 9 * (eta(q^9) / eta(q))^3 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u * v - 1)^3 - (u^3 - 1) * (v^3 - 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u + 2)^3 - 9 * v^3 * (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) = (u1 + 2) * (u2 + 2) - 3 * (1 + u1 + u2) * u3*u6.
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 + 9 * x * (Product_{k>0} (1 - x^(9*k)) / (1 - x^k))^3.
a(n) = 9 * A121589(n) unless n=0.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (3 * sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Nov 14 2015