OFFSET
1,4
FORMULA
G.f. A(x) satisfies: Sum_{n>=1} A((x-x^2)^n) = x. - Paul D. Hanna, Jan 04 2015
a(n) = Sum_{d|n} Moebius(n/d) * binomial(2*(d-1), d-1)/d. - Paul D. Hanna, Jan 04 2015
a(n) ~ 2^(2*n-2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 11 2019
EXAMPLE
G.f. = x + x^3 + 4*x^4 + 13*x^5 + 40*x^6 + 131*x^7 + 424*x^8 + 1428*x^9 + ...
MATHEMATICA
Table[Sum[MoebiusMu[n/d]*CatalanNumber[d-1], {d, Divisors[n]}], {n, 1, 30}] (* Vaclav Kotesovec, Sep 10 2019 *)
PROG
(PARI) /* Dirichlet convolution of mu(n) with Catalan numbers: */
{a(n) = sumdiv(n, d, moebius(n/d) * binomial(2*(d-1), d-1)/d)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 04 2015
(PARI) /* G.f. satisfies: Sum_{n>=1} A((x-x^2)^n) = x: */
{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-x^2 +x*O(x^#A))^n)))[#A]); A[n]}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 04 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved