0,2

Expansion of q^(-1/2) * (theta_2(q)^2 + 3 * theta_2(q^3)^2) / 4 in powers of q.

Expansion of phi(q) * psi(q) * psi(q^3) / phi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.

Euler transform of period 12 sequence [ 3, -4, 2, -2, 3, -2, 3, -2, 2, -4, 3, -2, ...].

Moebius transform is period 24 sequence [ 1, -1, 2, 0, 1, -2, -1, 0, -2, -1, -1, 0, 1, 1, 2, 0, 1, 2, -1, 0, -2, 1, -1, 0, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1 + (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1+(-1)^e)/2 if p = 7, 11 (mod 12).

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 6 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A132003.

a(6*n + 3) = a(6*n + 5) = 0.

q + 3*q^3 + 2*q^5 + q^9 + 2*q^13 + 6*q^15 + 2*q^17 + 3*q^25 + 3*q^27 + ...

(PARI) {a(n) = if( n<0 | n%6==3 | n%6==5, 0, if( n%2, 3, 1) * sumdiv( (n \ if( n%2, 3, 1)) * 2 + 1, d, kronecker( -4, d) ) )}

(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p==3, 2 - (-1)^e, if(p%12<6, e+1, (1 + (-1)^e) / 2))))))}

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)^3), n))}

A116604(n) = (-1)^n * a(n). A008441(n) = a(2*n). A002175(n) = a(6*n). 3 * A008441(n) = a(6*n + 1). 2 * A121444(n) = a(6*n + 2).

Sequence in context: A133209 A131290 A116604 this_sequence A079618 A117406 A008783

Adjacent sequences: A138738 A138739 A138740 this_sequence A138742 A138743 A138744

nonn

Michael Somos, Mar 27 2008