A045833 Expansion of eta(q^9)^3 / eta(q^3) in powers of q.
0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0
Offset: 0
Examples
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + ...
Links
- Amiram Eldar, Table of n, a(n) for n = 0..10000
Programs
-
Mathematica
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^9]^3 / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, Feb 22 2015 *) f[p_, e_] := If[Mod[p, 3] == 1, e + 1, (1 + (-1)^e)/2]; f[3, e_] := 0; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Oct 13 2022 *) -
PARI
{a(n) = local(A, p, e); if( n<0, 0, A=factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p!=3, if( p%3==1, e+1, !(e%2))))))}; \\ Michael Somos, May 25 2005 -
PARI
{a(n) = local(A); if( (n<1) || (n%3!=1), 0, n = (n-1)/3; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; \\ Michael Somos, May 25 2005
Formula
From Michael Somos, May 25 2005: (Start)
Euler transform of period 9 sequence [ 0, 0, 1, 0, 0, 1, 0, 0, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3.
Multiplicative with a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(9*sqrt(3)) = 0.403066... . - Amiram Eldar, Oct 13 2022
Dirichlet g.f.: L(chi_1,s)*L(chi_{-1},s), where chi_1 = A011655 and chi_{-1} = A102283 are respectively the principal and the non-principal Dirichlet character modulo 3. For the formula of the sequence whose Dirichlet g.f. is Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k, see A378006. This sequence is the case k = 3. - Jianing Song, Nov 13 2024