%I #13 Mar 12 2021 22:24:47

%S 1,-2,0,0,2,0,0,0,0,-2,0,0,0,0,0,0,3,-2,0,0,2,0,0,0,0,-4,0,0,0,0,0,0,

%T 2,0,0,0,2,0,0,0,0,-2,0,0,0,0,0,0,1,-4,0,0,4,0,0,0,0,-2,0,0,0,0,0,0,4,

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

%N Expansion of q^2 * phi(-q) * psi(q^16) in powers of q 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="/A227395/b227395.txt">Table of n, a(n) for n = 2..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="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of eta(q)^2 * eta(q^32)^2 / (eta(q^2) * eta(q^16)) in powers of q.

%F Euler transform of period 32 sequence [ -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

%F a(4*n) = a(4*n + 1) = a(8*n + 7) = 0. a(4*n + 2) = A113411(n). a(8*n + 3) = -2 * A033761(n).

%F G.f.: x^2 * Product_{k>0} (1 - x^k)^2 * (1 - x^(32*k))^2 / ((1 - x^(2*k)) * (1 - x^(16*k))).

%F a(n) = (-1)^n * A255258(n). - _Michael Somos_, Feb 20 2015

%e G.f. = q^2 - 2*q^3 + 2*q^6 - 2*q^11 + 3*q^18 - 2*q^19 + 2*q^22 - 4*q^27 + 2*q^34 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^8] / 2, {q, 0, n}];

%o (PARI) {a(n) = local(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^32 + A)^2 / (eta(x^2 + A) * eta(x^16 + A)), n))};

%Y Cf. A033761, A113411, A255258.

%K sign

%O 2,2

%A _Michael Somos_, Jul 10 2013