A136028 Expansion of (phi(q) * phi(q^2))^3 in powers of q where phi() is a Ramanujan theta function.

1, 6, 18, 44, 90, 144, 212, 288, 330, 418, 528, 588, 836, 1008, 1056, 1440, 1386, 1356, 1894, 1644, 2064, 2880, 2484, 3168, 3428, 2838, 3696, 3864, 4128, 5040, 5280, 5760, 5418, 5656, 5988, 5376, 7678, 8208, 7572, 10080, 8208, 7788, 10560, 8652, 10404, 13104

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q^2) * eta(q^4))^9 / (eta(q) * eta(q^8))^6 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1/(8 t)) = 2^(9/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

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

a(n) = A029713(n) + 6 * A030207(n). Convolution of A033715 and A097057.

a(n) = A028578(4*n). - Michael Somos, Oct 14 2015

Example

G.f. = 1 + 6*q + 18*q^2 + 44*q^3 + 90*q^4 + 144*q^5 + 212*q^6 + 288*q^7 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[((1-x^k)^2 * (1+x^k)^4 * (1+x^(2*k)) / (1+x^(4*k))^2)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^3, {q, 0, n}]; (* Michael Somos, Oct 14 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A) * eta(x^4 + A))^3 / (eta(x + A) * eta(x^8 + A))^2)^3, n))};
(Magma) A := Basis( ModularForms( Gamma1(8), 3), 46);  A[1] + 6*A[2] + 18*A[3] + 44*A[4] + 90*A[5] + 144*A[6] + 212*A[7]; /* Michael Somos, Oct 14 2015 */

Crossrefs

Cf. A029713, A030207, A033715, A097057.

Sequence in context: A344992 A011929 A070735 * A083719 A182706 A303649

Adjacent sequences: A136025 A136026 A136027 * A136029 A136030 A136031

Keyword

nonn

Author

Michael Somos, Dec 10 2007

Status

approved