A128486 Expansion of ((b(q)*c(q))^3 - 8*(b(q^2)*c(q^2))^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.

1, -14, 9, 52, 6, -126, -40, 136, 81, -84, -564, 468, 638, 560, 54, -2480, 882, -1134, -556, 312, -360, 7896, -840, 1224, -3089, -8932, 729, -2080, 4638, -756, 4400, 10528, -5076, -12348, -240, 4212, -2410, 7784, 5742, 816, -6870, 5040, 9644, -29328, 486, 11760, -18672, -22320

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..48.

Formula

Expansion of b(q) * b(q^2) * c(q) * c(q^2) * (b(q) * b(q^2) - c(q) * c(q^2)) / 9 in powers of q where b(), c() are cubic AGM theta functions.

Expansion of (eta(q) * eta(q^3))^6 - 8*(eta(q^2) * eta(q^6))^6 in powers of q.

Expansion of eta(q) * eta(q^2) * eta(q^3) * eta(q^6) * ((eta(q) * eta(q^2))^4 - 9*(eta(q^3) * eta(q^6))^4) in powers of q.

G.f.: x * Product_{k>0} (1 - x^k)^6 * (1 - x^(3*k))^6 - 8 * x^2 * Product_{k>0} (1 - x^(2*k))^6 * (1 - x^(6*k))^6.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6*t)) = -216 (t/i)^6 f(t) where q = exp(2 Pi i t).

a(2*n) = A007332(2*n) - 8 * A007332(n). a(2*n + 1) = A007332(2*n + 1).

Example

G.f. = q - 14*q^2 + 9*q^3 + 52*q^4 + 6*q^5 - 126*q^6 - 40*q^7 + 136*q^8 + ...

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 - 8*x * (eta(x^2 + A) * eta(x^6+A))^6, n))};
(PARI) {a(n) = my(A, A1, A2); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A) * eta(x^2 + A); A2 = eta(x^3 + A) * eta(x^6 + A); polcoeff( A1^5*A2 - 9*x * A1*A2^5, n))};
(Magma) A := Basis( CuspForms( Gamma1(6), 6), 49); A[1] - 14*A[2] + 9*A[3]; /* Michael Somos, Feb 19 2015 */

Crossrefs

Cf. A007332.

Sequence in context: A206641 A266014 A086050 * A147370 A140739 A305946

Adjacent sequences: A128483 A128484 A128485 * A128487 A128488 A128489

Keyword

sign,mult

Author

Michael Somos, Mar 04 2007

Status

approved