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A143434 Expansion of f(x, -x^3) in powers of x where f(,) is Ramanujan's two-variable theta function.

%I A143434 #13 Jun 16 2017 19:27:49
%S A143434 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 A143434 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 A143434 -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 A143434 Expansion of f(x, -x^3) in powers of x where f(,) is Ramanujan's two-variable theta function.
%H A143434 G. C. Greubel, <a href="/A143434/b143434.txt">Table of n, a(n) for n = 0..1000</a>
%F A143434 Euler transform of period 16 sequence [ 1, -1, -1, 1, -1, 0, 1, -2, 1, 0, -1, 1, -1, -1, 1, -1, ...].
%F A143434 G.f.: Sum_{k>=0} (-1)^floor((k + 2) / 4) * x^(k * (k+1) / 2).
%F A143434 a(n) = (-1)^n * A143433(n).
%e A143434 1 + x - x^3 - x^6 - x^10 - x^15 + x^21 + x^28 + x^36 + x^45 - x^55 - x^66 + ...
%e A143434 q + q^9 - q^25 - q^49 - q^81 - q^121 + q^169 + q^225 + q^289 + q^361 + ...
%t A143434 a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), I x^2], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> I x}, {x, 0, n}]]
%o A143434 (PARI) {a(n) = if( n<0, 0, if( issquare( 8*n + 1, &n), n = n\2; (-1)^((n + 2) \ 4), 0))}
%o A143434 (PARI) {a(n) = local(A); if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^( [1, -1, 1, 1, -1, 1, 0, -1, 2, -1, 0, 1, -1, 1, 1, -1] [k%16 + 1]), 1 + x * O(x^n)), n))}
%Y A143434 Cf. A143433.
%K A143434 sign
%O A143434 0,1
%A A143434 _Michael Somos_, Aug 14 2008