1,5

phi(x)=theta_3(x)=Sum_{k} x^(k^2) is a Ramanujan theta function.

Expansion of (eta(q^2)^3*eta(q^6)*eta(q^12)^2)/(eta(q)*eta(q^3)*eta(q^4)^2) in powers of q.

Euler transform of period 12 sequence [1, -2, 2, 0, 1, -2, 1, 0, 2, -2, 1, -2, ...].

Moebius transform is period 12 sequence [1, 0, -2, 0, 1, 0, -1, 0, 2, 0, -1, 0, ...].

a(n) is multiplicative and a(2^e) = 1, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).

G.f.: ((Sum_{k} x^(k^2))^2-(Sum_{k} x^(3k^2))^2)/4.

G.f.: Sum_{k>0} x^(3k-1)/(1+x^(6k-2)) +x^(3k-2)/(1+x^(6k-4)).

G.f.: Sum_{k>0} x^k(1-x^(2k))^2/(1+x^(6k)).

G.f.: x*Product_{k>0} (1-x^k)^2*(1+x^k)^3*(1+x^(3k))*(1+x^(4k)+x^(8k))^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A138949.

(PARI) a(n)=if(n<1, 0, (-1)^valuation(n, 3)*sumdiv(n, d, kronecker(-36, d)))

(PARI) {a(n)=if(n<1, 0, direuler(p=2, n, if(p==3, 1/(1+X), 1/(1-X)/(1-kronecker(-36, p)*X)))[n])}

(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x^2+A)^3*eta(x^6+A)*eta(x^12+A)^2/(eta(x+A)*eta(x^3+A)*eta(x^4+A)^2), n))}

Cf. a(n)=(-1)^e*A035154(n) where 3^e is the highest power of 3 dividing n.

A008441(n)=a(4n+1).

Sequence in context: A035188 A066295 A035154 this_sequence A121450 A132004 A109294

Adjacent sequences: A113443 A113444 A113445 this_sequence A113447 A113448 A113449

sign,mult

Michael Somos, Nov 02 2005