0,10

I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

a(n) is multiplicative and a(2^e) = -(1+(-1)^e)/2, if e>0, a(3^e) = -2(1+(-1)^e)/2 if e>0, a(p^e) = (1+(-1)^e)/2 otherwise.

Euler transform of period 6 sequence [1, -1, 0, -1, 1, -1, ...]. - Michael Somos Nov 05 2005

G.f.: (Sum_{k} 3(-x)^((3k)^2) - (-x)^(k^2))/2 = Product_{k>0} (1-x^(2k))/((1-x^(6k-1))(1-x^(6k-5))) - Michael Somos Nov 05 2005

Expansion of eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos Nov 05 2005

Expansion of psi(q) * chi(-q^3) in powers of q where psi(), chi() are Ramanujan theta functions. - Michael Somos Sep 16 2007

Expansion of (3 * phi(-q^9) - phi(-q)) / 2 in powers of q where phi() is a Ramanujan theta function.

Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q. - Michael Somos Sep 17 2007

Expansion of f(x*w, x/w) in powers of x where w is a primitive sixth root of unity and f() is Ramanujan's two variable theta function. - Michael Somos Sep 17 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 72^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A080995. - Michael Somos Jan 26 2008

G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^k + x^(2*k)). - Michael Somos Jan 26 2008

1 + q - q^4 - 2*q^9 - q^16 + q^25 + 2*q^36 + q^49 - q^64 - 2*q^81 + ...

(PARI) {a(n) = local(x); if( n<1, n==0, issquare(n, &x) * (1 + (n%3==0)) * (-1)^((1 + x) \ 3))} /* Michael Somos Nov 05 2005 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) / ( eta(x + A) * eta(x^6 + A) ), n))} /* Michael Soos Jan 26 2008 */

Sequence in context: A060478 A088806 A089807 this_sequence A096562 A096563 A078359

Adjacent sequences: A089807 A089808 A089809 this_sequence A089811 A089812 A089813

sign,mult

Eric Weisstein (eric(AT)weisstein.com), Nov 12, 2003