1,8

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(v).

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^6)^2 * eta(q^10)^2 * eta(q^30) / (eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20)) in powers of q.

Euler transform of period 60 sequence [ 0, -1, 1, -1, 1, -2, 0, -1, 1, -2, 0, -1, 0, -1, 2, -1, 0, -2, 0, -1, 1, -1, 0, -1, 1, -1, 1, -1, 0, -4, 0, -1, 1, -1, 1, -1, 0, -1, 1, -1, 0, -2, 0, -1, 2, -1, 0, -1, 0, -2, 1, -1, 0, -2, 1, -1, 1, -1, 0, -2, ...].

a(n) is multiplicative with a(2^e) = (-1)^e * (e-1) 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 / (60 t)) = 60^(1/2) (t/i) f(t) where q = exp(2 pi i t).

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)^(n + 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 * (e-1), 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^6 + A)^2 * eta(x^10 + A)^2 * eta(x^30 + A) / (eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A)), n))}

-(-1)^n * A140728(n) = a(n). A122855(n) = |a(n)|.

Sequence in context: A000161 A060398 A122855 this_sequence A140728 A130068 A051699

Adjacent sequences: A140724 A140725 A140726 this_sequence A140728 A140729 A140730

sign,mult

Michael Somos, May 29 2008