A113448 Expansion of (eta(q^2)^2 * eta(q^9) * eta(q^18)) / (eta(q) * eta(q^6)) in powers of q.
1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 3, 0, 1, 0, 0, 2, 2, 0
Offset: 1
Examples
G.f. = x + x^2 + x^4 + 2*x^7 + x^8 + 2*x^13 + 2*x^14 + x^16 + 2*x^19 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
Programs
-
Mathematica
a[ n_] := If[ n < 1, 0, If[ Mod[n, 3] == 0, 0, DivisorSum[ n, KroneckerSymbol[ -12, #] &]]]; (* Michael Somos, Jul 30 2015 *) a[ n_] := SeriesCoefficient[ x QPochhammer[ x^9]^3 / QPochhammer[ x^3] + x^2 QPochhammer[ x^18]^3 / QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jul 30 2015 *)
-
PARI
{a(n) = if( n<1, 0, if( n%3, sumdiv(n,d, kronecker(-12, d))))}; -
PARI
{a(n) = if( n<1, 0, direuler(p=2, n, if( p==3, 1, 1 / ((1 - X) * (1 - kronecker(-12, p)*X))))[n])} -
PARI
{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 0, p%6==1, e+1, !(e%2))))}; -
PARI
{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^9 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A)), n))}; -
PARI
{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) + x * eta(x^18 + A)^3 / eta(x^6 + A), n))};
Formula
Euler transform of period 18 sequence [ 1, -1, 1, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 1, -1, 1, -2, ...].
Moebius transform is period 18 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, 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(3*n) = 0, a(2*n) = a(n).
G.f.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - x^(18*k - 15) / (1 - x^(18*k - 15)) + x^(18*k - 6) / (1 - x^(18*k - 6)).
G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) * (1 - x^(4*k)) * (1-x^(10*k)) / (1 - x^(18*k)).
Expansion of (c(q) + c(q^2))/3 in powers of q^(1/3) where c(q) is a cubic AGM theta function.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Oct 15 2022
Comments