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%I #14 Mar 12 2021 22:24:45
%S 1,-1,0,1,0,0,-1,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,
%T 0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
%U -1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Expansion of f(-x, x^3) in powers of x where f(,) is Ramanujan's general 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="/A143433/b143433.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, 0, 1, 1, 1, -1, -1, -2, -1, -1, 1, 1, 1, 0, -1, -1, ...].
%F Pattern of signs of nonzero terms is A143431.
%F G.f.: Sum_{k>=0} (-1)^(k + floor(k/4)) * x^(k * (k+1) / 2).
%F a(n) = (-1)^n * A143434(n).
%F a(2*n) = A244465(n). a(2*n + 1) = - A244525(n). a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = 0.
%e G.f. = 1 - x + x^3 - x^6 - x^10 + x^15 - x^21 + x^28 + x^36 - x^45 + x^55 + ...
%e G.f. = q - q^9 + q^25 - q^49 - q^81 + q^121 - q^169 + q^225 + q^289 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x^4] QPochhammer[ -x^3, -x^4] QPochhammer[ -x^4], {x, 0, n}]; (* _Michael Somos_, Jun 03 2015 *)
%o (PARI) {a(n) = if( n<0, 0, if( issquare(8*n + 1, &n), n = n\2; (-1)^(n + n\4), 0))};
%o (PARI) {a(n) = my(A); if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^( [1, 1, 0, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, 0, 1, 1] [k%16 + 1]), 1 + x * O(x^n)), n))};
%Y Cf. A143434, A244465, A244525.
%K sign
%O 0,1
%A _Michael Somos_, Aug 14 2008
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