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A214303 Expansion of f(-x^2, -x^4) * f(x^1, x^7) in powers of x where f(,) is Ramanujan's two-variable theta function.

%I A214303 #13 Feb 16 2025 08:33:18
%S A214303 1,1,-1,-1,-1,-1,0,1,0,-1,2,0,-1,0,0,1,0,1,0,0,1,1,1,0,-1,-1,-1,1,0,
%T A214303 -1,-1,-3,1,0,-1,0,1,0,0,0,-1,1,0,0,1,2,-1,-1,0,-1,0,0,1,1,0,1,-1,-1,
%U A214303 0,2,0,0,2,0,0,0,2,0,0,-1,-1,0,0,0,1,-1,0,-1,-1
%N A214303 Expansion of f(-x^2, -x^4) * f(x^1, x^7) in powers of x where f(,) is Ramanujan's two-variable theta function.
%C A214303 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A214303 a(n) = sum of (-1)^((u-1)/6) over all solutions of 48*n + 31 = 4*u^2 + 3*v^2 in integers where u == 1 (mod 6) and v == 3 (mod 8).
%H A214303 G. C. Greubel, <a href="/A214303/b214303.txt">Table of n, a(n) for n = 0..1000</a>
%H A214303 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214303 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214303 Euler transform of period 16 sequence [ 1, -2, 0, -1, 0, -1, 1, -2, 1, -1, 0, -1, 0, -2, 1, -2, ...].
%F A214303 G.f.: (Sum_{k} (-1)^k * x^(3*k^2 + k)) * (Sum_{k} x^(4*k^2 + 3*k)).
%F A214303 a(n) = - A143379(2*n+1).
%e A214303 1 + x - x^2 - x^3 - x^4 - x^5 + x^7 - x^9 + 2*x^10 - x^12 + x^15 + x^17 + ...
%e A214303 q^31 + q^79 - q^127 - q^175 - q^223 - q^271 + q^367 - q^463 + 2*q^511 + ...
%t A214303 a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^8] QPochhammer[ -q^1, q^8] QPochhammer[ -q^7, q^8], {q, 0, n}]
%o A214303 (PARI) {a(n) = local(s, v); if( n<0, 0, n = 48*n + 31; forstep( u=1, sqrtint( n\4), 2, if( u%3 && issquare( (n - 4*u^2)/3, &v), s += (-1)^((u+1)\6))); s)}
%Y A214303 Cf. A010815, A143379, A214263.
%K A214303 sign
%O A214303 0,11
%A A214303 _Michael Somos_, Jul 11 2012