OFFSET
1,1
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001), 131-139. See p. 133 Theorem 2.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of (- phi(-q) + phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
Expansion of x * f(-x^2, -x^2) * f(x^2, x^14) / f(x, x^3) in powers of x where f() is Ramanujan's two-variable theta function.
a(n) is multiplicative with a(2) = -1, a(2^e) = (-1)^(e+1) if e>1, a(p^e) = (1 + (-1)^e) / 2 if p>2.
Euler transform of period 32 sequence [ -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, ...].
G.f.: (theta_4(q^2) - theta_4(q)) / 2 = (Sum_{k>0} (-1)^k * (x^(2*k^2) - x^(k^2))).
a(n) = -(-1)^(n * (n + 1)/2) * A143259(n).
Dirichlet g.f.: (1 - 1/2^(2*s-1)) * (1 - 1/2^s) * zeta(2*s). - Amiram Eldar, Sep 12 2023
EXAMPLE
G.f. = q - q^2 - q^4 + q^8 + q^9 - q^16 - q^18 + q^25 + q^32 - q^36 + q^49 - q^50 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] - EllipticTheta[ 4, 0, q]) / 2, {q, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^(n (n + 1)/2) ( Boole @ OddQ[ Length @ Divisors[ 2 n]] - Boole @ OddQ[ Length @ Divisors[ n]])];
PROG
(PARI) {a(n) = (-1)^(n * (n + 1)/2) * (issquare(2*n) - issquare(n))};
CROSSREFS
KEYWORD
sign,easy,mult
AUTHOR
Michael Somos, Jul 19 2012
STATUS
approved