A136747 Expansion of a(q)^2 * (b(q) * c(q) / 3)^3 in powers of q where a(), b(), c() are cubic AGM theta functions.

1, 6, -27, -92, 390, -162, -64, -1320, 729, 2340, -948, 2484, -5098, -384, -10530, 3856, 28386, 4374, -8620, -35880, 1728, -5688, -15288, 35640, 73975, -30588, -19683, 5888, 36510, -63180, -276808, 192096, 25596, 170316, -24960, -67068, 268526, -51720, 137646

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..39.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.] See page 30, line -3.

LMFDB, Newforms of weight 8 on Gamma_0(3).

Formula

Expansion of (eta(q) * eta(q^3))^6 * ((eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3))^2 in powers of q.

a(n) is multiplicative with a(3^e) = (-27)^e, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) unless p = 3.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^4*w + 512*u^3*w^2 + 131072*u^2*w^3 + 16777216*u*w^4 - 24*u^3*v*w - 9216*u^2*v*w^2 - 1572864*u*v*w^3 + 288*u^2*v^2*w + 73728*u*v^2*w^2 - u^2*v^3 - 1984*w*v^3*u - 65536*w^2*v^3 + 12*v^4*u + 3072*w*v^4 - 36*v^5.

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 Pi i t).

G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^6 * (Sum_{j,k in Z} x^(j*j + j*k + k*k))^2.

Convolution of A007332 and A008653.

Example

G.f. = q + 6*q^2 - 27*q^3 - 92*q^4 + 390*q^5 - 162*q^6 - 64*q^7 - 1320*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^3])^6 ((QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3])^2, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6 * sum(k=1, n, 12 * (sigma(3*k) - 3 * sigma(k)) * x^k, 1 + A), n))};
(Sage) CuspForms( Gamma0(3), 8, prec=40).0; # Michael Somos, May 28 2013
(Magma) Basis( CuspForms( Gamma0(3), 8), 40)[1]; /* Michael Somos, Oct 12 2015 */

Crossrefs

Cf. A007332, A008653.

Sequence in context: A264026 A341385 A344100 * A278357 A001940 A320049

Adjacent sequences: A136744 A136745 A136746 * A136748 A136749 A136750

Keyword

sign,mult

Author

Michael Somos, Jan 20 2008

Status

approved