%I #9 Mar 12 2021 22:24:48

%S 1,3,1,-2,3,4,-3,-3,2,7,0,-9,4,9,-5,-11,6,18,-7,-18,9,20,-12,-27,14,

%T 36,-11,-42,18,46,-24,-54,23,69,-27,-79,37,90,-44,-104,48,126,-52,

%U -147,65,162,-78,-189,85,225,-91,-254,114,286,-136,-327,142,381,-159

%N Expansion of phi(x)^2 / psi(x) 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="/A260313/b260313.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 f(-x) * chi(x)^4 = phi(-x^2) * chi(x)^3 = psi(x)^3 / psi(x^2)^2 = phi(-x^2)^4 / f(-x)^3 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

%F Expansion of q^(1/8) * eta(q^2)^8 / (eta(q)^3 * eta(q^4)^4) in powers of q.

%F Euler transform of period 4 sequence [ 3, -5, 3, -1, ...].

%F a(n) = A153172(n) + 2*A153174(n).

%e G.f. = 1 + 3*x + x^2 - 2*x^3 + 3*x^4 + 4*x^5 - 3*x^6 - 3*x^7 + 2*x^8 + ...

%e G.f. = 1/q + 3*q^7 + q^15 - 2*q^23 + 3*q^31 + 4*q^39 - 3*q^47 - 3*q^55 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 2 x^(1/8) / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / (eta(x + A)^3 * eta(x^4 + A)^4), n))};

%Y Cf. A153172, A153174.

%K sign

%O 0,2

%A _Michael Somos_, Jul 22 2015