0,9

A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms

Expansion of (eta(q^2)^2*eta(q^6)eta(q^10)eta(q^30)^2)/(eta(q)eta(q^4)eta(q^15)eta(q^60)) in powers of q.

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

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

Moebius transform is period 60 sequence [ 1, -1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, -1, -1, 0, 1, 1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, -1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 1, -1, 0, ...].

a(15n+7)=a(15n+11)=a(15n+13)=a(15n+14)=0. a(3n)=a(5n)=a(n).

G.f.: 1+Sum_{k>0} kronecker(-15,k) x^k/(1-(-x)^k).

(PARI) {a(n)=if(n<1, n==0, sumdiv(n, d, kronecker(-15, d)*(-1)^(d%4==2)))}

(PARI) {a(n)= local(A, p, e); if(n<1, n==0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, e-1, if(p<7, 1, if(p%15==2^valuation(p%15, 2), e+1, 1-e%2))))))}

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^6+A)*eta(x^10+A)*eta(x^30+A)^2/ (eta(x+A)*eta(x^4+A)*eta(x^15+A)*eta(x^60+A)), n))}

A035175(n)=a(4n).

Sequence in context: A086017 A000161 A060398 this_sequence A140727 A140728 A130068

Adjacent sequences: A122852 A122853 A122854 this_sequence A122856 A122857 A122858

nonn,mult

Michael Somos, Sep 14 2006