1,8

Expansion of q * f(-q^2) * f(-q^30) * chi(-q^3) * chi(-q^5) in powers of q where f(), chi() are Ramanujan theta functions.

Expansion of eta(q^2) * eta(q^3) * eta(q^5) * eta(q^30) / (eta(q^6) * eta(q^10)) in powers of q.

Euler transform of period 30 sequence [ 0, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, -2, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -2, ...].

a(n) is multiplicative with a(2^e) = (-1)^e * (1-e) if e>0. a(3^e) = a(5^e) = (-1)^e, a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (-1)^e * (e+1) if p == 2, 8 (mod 15), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15).

G.f. is a period 1 Fourier series which satisfies f( -1 / (30 t)) = 60^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A121362.

G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(30*k)) / ((1 + x^(3*k)) * (1 + x^(5*k))).

G.f.: Sum_{k>0} kronecker(5, n) * x^n / (1 - x^n + x^(2*n)) = Sum_{k>0} -(-1)^n * kronecker(5, n) * x^n / (1 + x^n + x^(2*n)).

-(-1)^n * A140727(n) = a(n). A122855(n) = |a(n)|

q - q^3 - q^4 - q^5 + 2*q^8 + q^9 + q^12 + q^15 - 3*q^16 - 2*q^17 + ...

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, -(-1)^d * kronecker(5, d) * kronecker(-3, n/d)))}

(PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if( p==2, (-1)^e * (1-e), if( p==3 | p==5, (-1)^e, if( kronecker(p, 15)==1, (e+1) * (-1)^(e*valuation(p%15, 2)), (1 + (-1)^e) / 2))))))}

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A) / (eta(x^6 + A) * eta(x^10 + A)), n))}

Sequence in context: A060398 A122855 A140727 this_sequence A130068 A051699 A007920

Adjacent sequences: A140725 A140726 A140727 this_sequence A140729 A140730 A140731

sign,mult

Michael Somos, May 29 2008