A244465 - OEIS

%I #10 Mar 12 2021 22:24:47

%S 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 -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 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

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

%H G. C. Greubel, <a href="/A244465/b244465.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 8 sequence [ 0, 0, -1, 0, -1, 0, 0, -1, ...].

%F G.f.: f(-x^3, -x^5) = Sum_{k in Z} (-1)^k * x^(4*k^2 - k).

%F a(n) = (-1)^n * A214264(n).

%e G.f. = 1 - x^3 - x^5 + x^14 + x^18 - x^33 - x^39 + x^60 + x^68 - x^95 + ...

%e G.f. = q - q^49 - q^81 + q^225 + q^289 - q^529 - q^625 + q^961 + q^1089 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^8] QPochhammer[ x^5, x^8] QPochhammer[ x^8], {x, 0, n}];

%o (PARI) {a(n) = issquare( 16*n + 1) * (-1)^n};

%Y Cf. A214264.

%K sign

%O 0,1

%A _Michael Somos_, Jun 28 2014