A033765 Product t2(q^d); d | 6, where t2 = theta2(q)/(2*q^(1/4)).

1, 1, 1, 3, 1, 2, 5, 2, 3, 7, 4, 4, 10, 3, 3, 11, 6, 4, 12, 6, 5, 19, 6, 8, 16, 7, 10, 17, 7, 8, 25, 10, 9, 20, 8, 8, 27, 12, 11, 30, 11, 14, 27, 12, 14, 29, 14, 12, 37, 15, 11, 42, 15, 14, 34, 12, 16, 44, 18, 16, 36, 18, 17, 39, 17, 20, 59, 18, 19, 42, 22, 24, 49

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Quadratic AGM theta functions: a(q) (see A004018), b(q) (A104794), c(q) (A005883).

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

Also the number of positive odd solutions to equation a^2 + 2*b^2 + 3*c^2 + 6*d^2 = 8*n + 12. - Seiichi Manyama, May 29 2017

Examples

			G.f. = 1 + x + x^2 + 3*x^3 + x^4 + 2*x^5 + 5*x^6 + 2*x^7 + 3*x^8 + 7*x^9 + ...
G.f. = q^3 + q^5 + q^7 + 3*q^9 + q^11 + 2*q^13 + 5*q^15 + 2*q^17 + 3*q^19 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Magma

A := Basis( ModularForms( Gamma0(24), 2), 105); A[4] + A[6]; /* Michael Somos, Aug 24 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^2] EllipticTheta[ 2, 0, q^3] EllipticTheta[ 2, 0, q^6] / 16, {q, 0, 2 n + 3}]; (* Michael Somos, Sep 30 2013 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 30 2013 */

Formula

Expansion of q^(-3) * (a(q) - a(q^3)) * c(q) / 16 in powers of q^2 where a(), c() are quadratic AGM theta functions. - Michael Somos, Sep 30 2013

Expansion of (phi(x)^2 - phi(x^3)^2) * psi(x^2)^2 / 4 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Sep 30 2013

Extensions

More terms from Seiichi Manyama, May 22 2017