A215711 Expansion of a(q) * b(q)^3 in powers of q where a(), b() are cubic AGM theta functions.

1, -3, -27, 159, -219, -378, 1431, -1032, -1755, 4533, -3402, -3996, 11607, -6594, -9288, 20034, -14043, -14742, 40797, -20580, -27594, 54696, -35964, -36504, 93015, -47253, -59346, 122631, -75336, -73170, 180306, -89376, -112347, 211788, -132678, -130032

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010. See Table 1, p.12

Formula

Expansion of (1 + 9 * q * (eta(q^9) / eta(q))^3) * (eta(q)^3 / eta(q^3))^4 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^5 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A198956.

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

Convolution of A215690 and A133078. Convolution of A004016 and A109041.

Example

G.f. = 1 - 3*q - 27*q^2 + 159*q^3 - 219*q^4 - 378*q^5 + 1431*q^6 - 1032*q^7 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(1 + 9*(eta[q^9]/eta[q])^3)*(eta[q]^3/eta[q^3])^4, {q, 0, 50}], q] (* G. C. Greubel, Aug 10 2018 *)

Prog

(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) * (eta(x + A)^3 / eta(x^3 + A))^4, n))}

Crossrefs

Cf. A004016, A109041, A133078, A198956, A215690.

Sequence in context: A108142 A056263 A026093 * A270956 A129530 A056370

Adjacent sequences: A215708 A215709 A215710 * A215712 A215713 A215714

Keyword

sign

Author

Michael Somos, Aug 21 2012

Status

approved