A259896 Expansion of psi(x) * psi(x^6) in powers of x where psi() is a Ramanujan theta function.

1, 1, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 3, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 2, 1, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0

Offset: 0

Author

Michael Somos, Jul 07 2015

Keywords

• nonn

Comments

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 x^2 + 6*y^2 = 8*n + 7. - Seiichi Manyama, May 28 2017

Examples

			G.f. = 1 + x + x^3 + 2*x^6 + x^7 + x^9 + x^10 + x^12 + x^15 + x^16 + ...
G.f. = q^7 + q^15 + q^31 + 2*q^55 + q^63 + q^79 + q^87 + q^103 + q^127 + ...
		

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

Crossrefs

Cf. A259895.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^3] / (4 q^(7/8)), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))};

Formula

Expansion of q^(-7/8) * eta(q^2)^2 * eta(q^12)^2 / (eta(q) * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ 1, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -2, ...].

a(3*n + 1) = A259895(n). a(3*n + 2) = a(9*n + 4) = 0.