A033764 Product t2(q^d); d | 5, where t2 = theta2(q)/(2*q^(1/4)).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Also the number of positive odd solutions to equation x^2 + 5*y^2 = 8*n + 6. - Seiichi Manyama, May 28 2017

Examples

			q^3 + q^7 + q^15 + q^23 + 2*q^27 + q^35 + q^43 + q^47 + 3*q^63 + q^67 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• A. Berkovich and H. Yesilyurt, Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms, arXiv:math/0611300 [math.NT], 2006-2007, see Equation (3.19).
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Mathematica

QP = QPochhammer; s = QP[q^2]^2*(QP[q^10]^2/(QP[q]*QP[q^5])) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^10 + A)^2 / eta(x + A) / eta(x^5 + A), n))} /* Michael Somos, Mar 21 2008 */

Formula

Expansion of psi(q) * psi(q^5) where psi() is a Ramanujan theta function.

Expansion of q^(-3/4) * eta(q^2)^2 * eta(q^10)^2 / (eta(q) * eta(q^5)) in powers of q.

Euler transform of period 10 sequence [ 1, -1, 1, -1, 2, -1, 1, -1, 1, -2, ...]. - Michael Somos, Mar 21 2008

G.f.: Sum_{k} (x^(3*k) + x^(7*k+1)) / (1 - x^(20*k+5)) = Sum_{k} (x^k + x^(9*k+6)) / (1 - x^(20*k+15)).