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%I #29 Mar 12 2021 22:24:42
%S 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,
%T 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,
%U 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
%N Product t2(q^d); d | 5, where t2 = theta2(q)/(2*q^(1/4)).
%C 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).
%C Also the number of positive odd solutions to equation x^2 + 5*y^2 = 8*n + 6. - _Seiichi Manyama_, May 28 2017
%H Seiichi Manyama, <a href="/A033764/b033764.txt">Table of n, a(n) for n = 0..10000</a>
%H A. Berkovich and H. Yesilyurt, <a href="http://arxiv.org/abs/math/0611300">Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms</a>, arXiv:math/0611300 [math.NT], 2006-2007, see Equation (3.19).
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of psi(q) * psi(q^5) where psi() is a Ramanujan theta function.
%F Expansion of q^(-3/4) * eta(q^2)^2 * eta(q^10)^2 / (eta(q) * eta(q^5)) in powers of q.
%F Euler transform of period 10 sequence [ 1, -1, 1, -1, 2, -1, 1, -1, 1, -2, ...]. - _Michael Somos_, Mar 21 2008
%F 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)).
%e q^3 + q^7 + q^15 + q^23 + 2*q^27 + q^35 + q^43 + q^47 + 3*q^63 + q^67 + ...
%t 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 *)
%o (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 */
%K nonn
%O 0,7
%A _N. J. A. Sloane_