A242042 Expansion of (b(q) * c(q^3) / 3)^2 in powers of q where b(), c() are cubic AGM theta functions.

1, -6, 9, 14, -54, 36, 65, -162, 126, 148, -438, 252, 344, -756, 513, 546, -1458, 756, 1022, -2064, 1332, 1352, -3510, 1764, 2198, -4374, 2808, 2710, -6804, 3276, 4161, -7992, 4914, 4816, -11826, 5616, 6860, -13188, 8190, 7658, -18576, 8892, 10804, -20412

Offset

2,2

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

In McKay and Sebbar on page 274 in equation (8.2) the last term on the right side is a multiple of the g.f.

Links

G. C. Greubel, Table of n, a(n) for n = 2..2500

J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann. 318 (2000), no. 2, 255-275. MR1795562 (2001m:11063)

Formula

Expansion of (eta(q) * eta(q^9))^6 / eta(q^3)^4 in powers of q.

Euler transform of period 9 sequence [ -6, -6, -2, -6, -6, -2, -6, -6, -8, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^2 f(t) where q = exp(2 Pi i t).

G.f.: x^2 * Product_{k>0} (1 - x^k)^6 * (1 - x^(9*k))^6 / (1 - x^(3*k))^4.

Convolution square of A106401.

a(3*n) = -6 * A198956(n). a(3*n + 1) = 9 * A033690(n).

Example

G.f. = q^2 - 6*q^3 + 9*q^4 + 14*q^5 - 54*q^6 + 36*q^7 + 65*q^8 - 162*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q] QPochhammer[ q^9])^6 / QPochhammer[ q^3]^4, {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A))^6 / eta(x^3 + A)^4, n))};
(Magma) A := Basis( ModularForms( Gamma0(9), 4), 19); A[3] - 6*A[4] + 9*A[5];

Crossrefs

Cf. A033680, A106401, A198956.

Sequence in context: A020717 A196993 A303162 * A316021 A185398 A316022

Adjacent sequences: A242039 A242040 A242041 * A242043 A242044 A242045

Keyword

sign

Author

Michael Somos, Aug 12 2014

Status

approved