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%I #10 Sep 08 2022 08:46:09
%S 3,2,-2,4,6,0,-4,0,6,6,0,4,12,0,0,0,6,4,-6,4,0,0,-4,0,12,2,0,8,0,0,0,
%T 0,6,8,-4,0,18,0,-4,0,0,4,0,4,12,0,0,0,12,2,-2,8,0,0,-8,0,0,8,0,4,0,0,
%U 0,0,6,0,-8,4,12,0,0,0,18,4,0,4,12,0,0,0,0
%N Expansion of phi(q) * phi(q^2) + 2 * phi(-q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A245572/b245572.txt">Table of n, a(n) for n = 0..2500</a>
%F a(n) = 2 * b(n) where b(n) is multiplicative with b(2) = -1, b(2^e) = 3 if e>1, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226240.
%F a(2*n + 1) = 2 * A113411(n). a(4*n) = 3 * A033715(n). a(8*n + 1) = 2 * A112603(n). a(8*n = 3) = 4 * A033761(n). a(8*n + 5) = a(8*n = 7) = 0.
%e G.f. = 3 + 2*q - 2*q^2 + 4*q^3 + 6*q^4 - 4*q^6 + 6*q^8 + 6*q^9 + 4*q^11 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] + 2 EllipticTheta[ 3, 0, -q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}];
%o (PARI) {a(n) = my(A, p, e); if( n<1, 3*(n==0), A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, if( e>1, 3, -1), p%8>3, (1 + (-1)^e) / 2, e+1)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^3 / (eta(x + A)^2 * eta(x^8 + A)^2) + 2 * eta(x^2 + A)^2 * eta(x^8 + A)^5 / (eta(x^4 + A)^3 * eta(x^16 + A)^2), n))};
%o (Magma) A := Basis( ModularForms( Gamma1(32), 1), 81); 3*A[1] + 2*A[2] - 2*A[3] + 4*A[4] + 6*A[5] - 4*A[7] + 6*A[9] + 6*A[10] + 4*A[12] + 12*A[13] + 4*A[16];
%Y Cf. A033715, A033761, A112603, A113411, A226240.
%K sign
%O 0,1
%A _Michael Somos_, Jul 25 2014
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