%I #16 Mar 12 2021 22:24:47

%S 1,-1,0,-1,0,0,-1,2,0,2,1,0,-2,0,0,-1,-2,0,0,0,0,1,0,0,2,-2,0,0,1,0,2,

%T 0,0,0,0,0,1,0,0,-2,0,0,-2,0,0,-3,0,0,0,0,0,2,2,0,-2,1,0,2,0,0,0,2,0,

%U 0,-2,0,1,0,0,0,0,0,-2,0,0,2,0,0,1,-2,0,0

%N Expansion of psi(-x) * phi(-x^6) in powers of x where phi(), psi() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A226860/b226860.txt">Table of n, a(n) for n = 0..2500</a>

%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 q^(-1/8) * eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^12)) in powers of q.

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 96^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226350.

%F a(3*n) = A226289(n). a(3*n + 1) = - A246962(n). a(3*n + 2) = 0.

%e G.f. = 1 - x - x^3 - x^6 + 2*x^7 + 2*x^9 + x^10 - 2*x^12 - x^15 - 2*x^16 + x^21 + ...

%e G.f. = q - q^9 - q^25 - q^49 + 2*q^57 + 2*q^73 + q^81 - 2*q^97 - q^121 - 2*q^129 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^6] EllipticTheta[ 2, Pi/4, q^(1/2)] / (Sqrt[2] q^(1/8)), {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^12 + A)), n))};

%o (PARI) {a(n) = my(A, p, e, i); if( n<0, 0, n = 8*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, I^e, p%24 == 1 || p%24==19, for(j=1, sqrtint(p\18), if( issquare( p - 18*j^2, &i), break)); (e+1) * (if(p%24==1, 1, -I) * kronecker( 12, i))^e, if( e%2, 0, if(p%24>12, 1, -1)^(e/2))) ))}; /* _Michael Somos_, Sep 08 2014 */

%Y Cf. A226350, A226289, A246962.

%K sign

%O 0,8

%A _Michael Somos_, Jun 20 2013