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%I #11 Mar 12 2021 22:24:46
%S 1,-1,-2,3,4,-6,-8,11,15,-20,-26,34,44,-56,-72,91,114,-143,-178,220,
%T 272,-334,-408,498,605,-732,-884,1064,1276,-1528,-1824,2171,2580,
%U -3058,-3616,4269,5028,-5910,-6936,8124,9498,-11088,-12922,15034,17468,-20264
%N Expansion of q * f(-q,-q^7)^2 / (phi(q^2) * psi(-q)) in powers of q where phi(), psi(), f(,) 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="/A224216/b224216.txt">Table of n, a(n) for n = 1..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 q * (f(-q^8) / f(q^2))^2 * f(-q,-q^7) / f(-q^3,-q^5) = q * f(-q,-q^7) * f(-q^2,-q^6)^2 / (f(-q^3,-q^5) * f(-q^4,-q^4)^2) in powers of q where f() is Ramanujan's theta function.
%F Euler transform of period 8 sequence [ -1, -2, 1, 4, 1, -2, -1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v) * (1 + 2*u) - (u + u^2) * (1 - 4*v - 4*v^2).
%F a(n) = (-1)^floor(n \ 2) * A115671(n) unless n=0.
%e q - q^2 - 2*q^3 + 3*q^4 + 4*q^5 - 6*q^6 - 8*q^7 + 11*q^8 + 15*q^9 + ...
%t a[ n_] := (-1)^Floor[ n / 2] SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] - 1) / 2, {q, 0, n}]
%t a[ n_] := SeriesCoefficient[ q^(1/2) QPochhammer[ -q] EllipticTheta[ 2, 0, q^2] / EllipticTheta[ 4, 0, q^4]^2 QPochhammer[ q, q^8]^2 QPochhammer[ q^7, q^8]^2 / 2, {q, 0, n}]
%o (PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); (-1)^(n \ 2) * polcoeff( (-1 + eta(x^2 + A)^3 / eta(x + A)^2 / eta(x^4 + A)) / 2, n))}
%Y Cf. A115671, A208856.
%K sign
%O 1,3
%A _Michael Somos_, Apr 01 2013
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