A214505 - OEIS

%I #15 Mar 12 2021 22:24:46

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

%T 0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,

%U 0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1

%N a(n) = 1 if n is four times a triangular number, -1 if one more than twelve times a triangular number else 0.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H Antti Karttunen, <a href="/A214505/b214505.txt">Table of n, a(n) for n = 0..65537</a>

%H S. Cooper and M. Hirschhorn, <a href="http://dx.doi.org/10.1216/rmjm/1008959672">On some infinite product identities</a>, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 5.

%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 Expansion of psi(x^4) - x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.

%F Expansion of f(-x, x^5) * f(-x^4, -x^8) / f(x, -x) in powers of x where f(,) is the Ramanujan two-variable theta function.

%F Euler transform of period 24 sequence [ -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, ...].

%F G.f.: (Sum_{k} x^(2*k*(k + 1)) - x^(6*k*(k + 1) + 1)) / 2.

%F a(n) = A214295(2*n + 1).

%e 1 - x + x^4 + x^12 - x^13 + x^24 - x^37 + x^40 + x^60 - x^73 + x^84 + ...

%e q - q^3 + q^9 + q^25 - q^27 + q^49 - q^75 + q^81 + q^121 - q^147 + q^169 + ...

%o (PARI) {a(n) = n = 2*n + 1; issquare(n) - issquare(3*n)}

%Y Cf. A010052, A214295.

%K sign

%O 0,1

%A _Michael Somos_, Jul 19 2012