OFFSET
1,3
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).
EXAMPLE
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) +...
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 04 2015
STATUS
approved