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A244465 Expansion of f(-x^3, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

%I A244465 #11 Feb 16 2025 08:33:23
%S A244465 1,0,0,-1,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A244465 -1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
%U A244465 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A244465 Expansion of f(-x^3, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.
%H A244465 G. C. Greubel, <a href="/A244465/b244465.txt">Table of n, a(n) for n = 0..1000</a>
%H A244465 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A244465 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A244465 Euler transform of period 8 sequence [ 0, 0, -1, 0, -1, 0, 0, -1, ...].
%F A244465 G.f.: f(-x^3, -x^5) = Sum_{k in Z} (-1)^k * x^(4*k^2 - k).
%F A244465 a(n) = (-1)^n * A214264(n).
%e A244465 G.f. = 1 - x^3 - x^5 + x^14 + x^18 - x^33 - x^39 + x^60 + x^68 - x^95 + ...
%e A244465 G.f. = q - q^49 - q^81 + q^225 + q^289 - q^529 - q^625 + q^961 + q^1089 + ...
%t A244465 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^8] QPochhammer[ x^5, x^8] QPochhammer[ x^8], {x, 0, n}];
%o A244465 (PARI) {a(n) = issquare( 16*n + 1) * (-1)^n};
%Y A244465 Cf. A214264.
%K A244465 sign
%O A244465 0,1
%A A244465 _Michael Somos_, Jun 28 2014