A251662 Dirichlet convolution of Moebius function mu(n) (A008683) with Ternary numbers A001764.

1, 0, 2, 11, 54, 270, 1427, 7740, 43260, 246620, 1430714, 8414356, 50067107, 300829144, 1822766463, 11124747912, 68328754958, 422030501802, 2619631042664, 16332922043614, 102240109896265, 642312449787030, 4048514844039119, 25594403732709300, 162250238001816845, 1031147983109715120

Offset: 1

Author

Paul D. Hanna, Jan 04 2015

Keywords

• nonn

Examples

			G.f.: A(x) = x + 2*x^3 + 11*x^4 + 54*x^5 + 270*x^6 + 1427*x^7 + 7740*x^8 +...
where Sum_{n>=1} A(x^n*(1-x)^(2*n)) = x - x^2:
x-x^2 = A(x*(1-x)^2) + A(x^2*(1-x)^4) + A(x^3*(1-x)^6) + A(x^4*(1-x)^8) +...
		

Crossrefs

Cf. A034742, A001764, A008683.

Programs

PARI

/* Dirichlet convolution of mu(n) with Ternary numbers A001764: */
{a(n) = sumdiv(n, d, moebius(n/d) * binomial(3*(d-1), d-1)/(2*d-1))}
for(n=1, 30, print1(a(n), ", "))

PARI

/* G.f. satisfies: Sum_{n>=1} A(x^n*(1-x)^(2*n)) = x-x^2. */
{a(n)=local(A=[1, 0]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(sum(n=1, #A, subst(x*Ser(A), x, (x-2*x^2+x^3 +x*O(x^#A))^n)))[#A]); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies: Sum_{n>=1} A((x - 2*x^2 + x^3)^n) = x - x^2.

a(n) = Sum_{d|n} Moebius(n/d) * binomial(3*(d-1), d-1)/(2*d-1).