%I #14 Feb 16 2025 08:33:23
%S 1,-2,0,0,0,4,0,0,-4,-2,0,0,0,4,0,0,4,-4,0,0,0,0,0,0,0,-6,0,0,0,4,0,0,
%T 4,0,0,0,0,4,0,0,-8,-4,0,0,0,4,0,0,0,-2,0,0,0,4,0,0,0,0,0,0,0,4,0,0,4,
%U -8,0,0,0,0,0,0,-4,-4,0,0,0,0,0,0,8,-2,0
%N Expansion of phi(-q) * 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="/A246950/b246950.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of f(-q, -q) * f(q, -q) in powers of q where f(,) is the Ramanujan general theta function. - _Michael Somos_, Jun 21 2015
%F Expansion of eta(q)^2 * eta(q^4)^2 / (eta(q^2) * eta(q^8)) in powers of q.
%F Euler transform of period 8 sequence [ -2, -1, -2, -3, -2, -1, -2, -2, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053692.
%F G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k)).
%F a(n) = (-1)^n * A204531(n). a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0.
%F a(8*n) = A104794(n). a(4*n + 1) = - 2 * A134343(n).
%F a(8*n + 1) = -2 * A113407(n). a(8*n + 5) = 4 * A053692(n). - _Michael Somos_, Jun 10 2015
%e G.f. = 1 - 2*q + 4*q^5 - 4*q^8 - 2*q^9 + 4*q^13 + 4*q^16 - 4*q^17 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^4], {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 / (eta(x^2 + A) * eta(x^8 + A)), n))};
%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * (-1)^(n%8==1) * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 2 * (e>2) * (-1)^(e<4), p%4==1, (e+1), !(e%2))))};
%o (Magma) A := Basis( ModularForms( Gamma1(64), 1), 85); A[1] - 2*A[2] + 4*A[6] - 4*A[9] - 2*A[10] + 4*A[14] + 4*A[17] - 4*A[18] - 6*A[26] + 4*A[30] - 4*A[35] + 4*A[36]; /* _Michael Somos_, Jun 21 2015 */
%Y Cf. A053692, A104794, A113407, A134343, A204531.
%K sign,changed
%O 0,2
%A _Michael Somos_, Sep 08 2014