A115235 Expansion of eta(q)^2*eta(q^9)*eta(q^18)/(eta(q^2)*eta(q^3)) in powers of q.

1, -2, 0, 1, 0, 0, 2, -2, 0, 0, 0, 0, 2, -4, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, -4, 0, 2, 0, 0, 2, -2, 0, 0, 0, 0, 2, -4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, -2, 0, 2, 0, 0, 0, -4, 0, 0, 0, 0, 2, -4, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, -4, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, -6, 0, 1, 0, 0, 2, -4, 0

Offset: 1

Author

Michael Somos, Jan 17 2006

Keywords

• sign
• easy
• mult

Examples

			q - 2*q^2 + q^4 + 2*q^7 - 2*q^8 + 2*q^13 - 4*q^14 + q^16 + 2*q^19 + ...
		

Crossrefs

Cf. A033687.

Programs

Mathematica

f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (-1+3*(-1)^e)/2; f[3, e_] := 0; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)

PARI

{a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^9+A)*eta(x^18+A)/eta(x^2+A)/eta(x^3+A), n))}

Formula

Euler transform of period 18 sequence [ -2, -1, -1, -1, -2, 0, -2, -1, -2, -1, -2, 0, -2, -1, -1, -1, -2, -2, ...].

Moebius transform is period 18 sequence [1, -3, -1, 3, -1, 3, 1, -3, 0, 3, -1, -3, 1, -3, 1, 3, -1, 0, ...].

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

a(3n+1) = A033687(n), a(6n+2) = -2*A033687(n), a(3n) = a(6n+5) = 0. a(4n) = a(n).

G.f.: Sum_{k} x^(3k+1)*(1-2*x^(3k+1))/(1-x^(18k+6)).