%I #32 May 06 2023 12:19:51
%S 1,-1,2,-1,0,2,-1,0,0,-2,-1,2,-2,0,-2,1,0,2,0,-2,0,1,0,0,-2,0,0,0,-1,
%T -2,2,0,2,0,0,2,3,0,0,-2,0,0,-2,0,2,-1,2,0,0,0,2,-2,0,-2,2,1,-2,2,0,0,
%U 0,0,0,0,0,2,-1,0,0,0,0,-2,2,0,-2,2,0,-2,-1,0,-2,0,-2,0,-2,0,0,4,0,0,0,1,0,0,0,-2,-2,0,0,0,2,2,0,0,-2
%N Expansion of f(-x, x^3) * phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Essentially the expansion of eta(q)*eta(q^2). Cf. A010815. - _N. J. A. Sloane_, Feb 18 2010
%C A030204, A083650 and A138514 are the same except for signs. - _N. J. A. Sloane_, May 07 2010
%H G. C. Greubel, <a href="/A083650/b083650.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="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Euler transform of period 16 sequence [ -1, 2, 1, -2, 1, 1, -1, -3, -1, 1, 1, -2, 1, 2, -1, -2, ...].
%F G.f.: (Sum_{k>=0} (-1)^(k + [k/4]) * x^(k*(k+1)/2)) * (Sum_k x^(2*k^2)).
%F (-1)^[n/2] * a(n) = A030204(n).
%e G.f. = 1 - x + 2*x^2 - x^3 + 2*x^5 - x^6 - 2*x^9 - x^10 + 2*x^11 - 2*x^12 - 2*x^14 + ...
%e G.f. = q - q^9 + 2*q^17 - q^25 + 2*q^41 - q^49 + 2*q^73 - q^81 + 2*q^89 - 2*q^97 + ...
%t QP = QPochhammer; s = QP[q]*QP[q^2] + O[q]^105; Table[(-1)^Quotient[n, 2]*Coefficient[s, q, n], {n, 0, 105}] (* _Jean-François Alcover_, Nov 27 2015, adapted from PARI *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); (-1)^(n\2) * polcoeff( eta(x + A) * eta(x^2 + A), n))}; /* _Michael Somos_, Mar 02 2010 */
%Y Cf. A000122, A000700, A010054, A010815, A030204, A083650, A121373, A138514, A143433.
%K sign
%O 0,3
%A _Michael Somos_, May 01 2003
%E Revised by _Michael Somos_, Mar 02 2010