A034742 Dirichlet convolution of Moebius function mu(n) (A008683) with Catalan numbers (A000108).

1, 0, 1, 4, 13, 40, 131, 424, 1428, 4848, 16795, 58740, 208011, 742768, 2674425, 9694416, 35357669, 129643320, 477638699, 1767258324, 6564120287, 24466250224, 91482563639, 343059554440, 1289904147310, 4861946193440, 18367353070722, 69533550173100, 263747951750359

Offset: 1

Author

Erich Friedman

Keywords

• nonn
• easy

Examples

			G.f. = x + x^3 + 4*x^4 + 13*x^5 + 40*x^6 + 131*x^7 + 424*x^8 + 1428*x^9 + ...
		

Programs

Mathematica

Table[Sum[MoebiusMu[n/d]*CatalanNumber[d-1], {d, Divisors[n]}], {n, 1, 30}] (* Vaclav Kotesovec, Sep 10 2019 *)

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

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