1,7

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

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

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

a(n) is multiplicative and a(2^e) = -(-1)^e if e>0, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).

G.f.: Sum_{k>0} x^(6k-5)/(1-x^(6k-5)) -x^(6k-1)/(1-x^(6k-1)) -2*x^(12k-8)/(1-x^(12k-8)) +2x^(12k-4)/(1-x^(12k-4)).

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

G.f.: x*Product_{k>0} (1-x^k)/(1-x^(4k-2))*((1-x^(12k-6))/(1-x^(3k)))^3.

Expansion of theta_2(iq^3)^3/(4*theta_2(iq)) in powers of q^2.

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

Expansion of (c(q) * c(q^4)) / (3 * c(q^2)) in powers of q where c() is a cubic AGM function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3)^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A132973.

a(3n) = a(n). a(6n+5) = a(12n+10) = 0.

q + q^2 + q^3 - q^4 + q^6 + 2*q^7 + q^8 + q^9 - q^12 + 2*q^13 + ...

(PARI) {a(n)=local(x); if(n<1, 0, x=valuation(n, 2); -(-1)^max(1, x)*sumdiv(n, d, kronecker(-12, d)))}

(PARI) {a(n)=if(n<1, 0, direuler(p=2, n, if(p==2, 1+X/(1+X), 1/(1-X)/(1-kronecker(-12, p)*X)))[n])}

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

Convolution inverse of A133637.

A035178(n) = |a(n)|. A093829(n) = a(2n). A033762(n) = a(2n+1). A112604(n) = a(4n+1). A112605(n) = a(4n+3). A097195(n) = a(6n+1). A033687(n) = a(6n+2).

A112606(n) = a(8n+1). A112608(n) = a(8n+3). 2 * A112607(n) = a(8n+5). A112605(n) = a(8n+6). 2 * A112609(n) = a(8n+7).

A123884(n) = a(12n+1). 2 * A121361(n) = a(12n+7). A131961(n) = a(24n+1). 2 * A131962(n) = a(24n+7). 2 * A131963(n) = a(24n+13). 2 * A131964(n) = a(24n+19).

Sequence in context: A061197 A035178 A093829 this_sequence A137608 A078807 A029422

Adjacent sequences: A113444 A113445 A113446 this_sequence A113448 A113449 A113450

sign,mult

Michael Somos, Nov 02 2005