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A214302 Expansion of f(-x^2, -x^4) * f(x^3, x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.

%I A214302 #10 Feb 16 2025 08:33:18
%S A214302 1,0,-1,1,-1,0,0,-2,0,-1,1,0,0,1,2,1,-1,1,0,1,-1,0,-1,0,0,0,0,-1,2,-1,
%T A214302 -1,0,1,0,0,-2,0,-1,-1,1,0,-1,-1,0,0,0,0,2,-1,2,0,0,1,1,0,1,0,0,1,0,1,
%U A214302 0,0,-2,-1,0,1,0,1,-1,0,0,-1,-1,1,-1,0,1,1
%N A214302 Expansion of f(-x^2, -x^4) * f(x^3, x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.
%C A214302 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A214302 a(n) = sum of (-1)^((u-1)/6) over all solutions of 48*n + 7 = 4*u^2 + 3*v^2 in integers where u == 1 (mod 6) and v == 1 (mod 8).
%H A214302 G. C. Greubel, <a href="/A214302/b214302.txt">Table of n, a(n) for n = 0..1000</a>
%H A214302 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214302 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214302 Euler transform of period 16 sequence [ 0, -1, 1, -1, 1, -2, 0, -2, 0, -2, 1, -1, 1, -1, 0, -2, ...].
%F A214302 G.f.: (Sum_{k} (-1)^k * x^(3*k^2 + k)) * (Sum_{k} x^(4*k^2 + k)).
%F A214302 a(n) = A143379(2*n).
%e A214302 1 - x^2 + x^3 - x^4 - 2*x^7 - x^9 + x^10 + x^13 + 2*x^14 + x^15 - x^16 + ...
%e A214302 q^7 - q^103 + q^151 - q^199 - 2*q^343 - q^439 + q^487 + q^631 + 2*q^679 + ...
%t A214302 a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^8] QPochhammer[ -q^3, q^8] QPochhammer[ -q^5, q^8], {q, 0, n}]
%o A214302 (PARI) {a(n) = local(s, v); if( n<0, 0, n = 48*n + 7; forstep( u=1, sqrtint( n\4), 2, if( u%3 && issquare( (n - 4*u^2)/3, &v), s += (-1)^((u+1)\6))); s)}
%Y A214302 Cf. A010815, A143379, A214264.
%K A214302 sign
%O A214302 0,8
%A A214302 _Michael Somos_, Jul 11 2012