|
%I #16 Sep 11 2019 05:08:26
%S 1,0,1,4,13,40,131,424,1428,4848,16795,58740,208011,742768,2674425,
%T 9694416,35357669,129643320,477638699,1767258324,6564120287,
%U 24466250224,91482563639,343059554440,1289904147310,4861946193440,18367353070722,69533550173100,263747951750359
%N Dirichlet convolution of Moebius function mu(n) (A008683) with Catalan numbers (A000108).
%F G.f. A(x) satisfies: Sum_{n>=1} A((x-x^2)^n) = x. - _Paul D. Hanna_, Jan 04 2015
%F a(n) = Sum_{d|n} Moebius(n/d) * binomial(2*(d-1), d-1)/d. - _Paul D. Hanna_, Jan 04 2015
%F a(n) ~ 2^(2*n-2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 11 2019
%e G.f. = x + x^3 + 4*x^4 + 13*x^5 + 40*x^6 + 131*x^7 + 424*x^8 + 1428*x^9 + ...
%t Table[Sum[MoebiusMu[n/d]*CatalanNumber[d-1], {d, Divisors[n]}], {n, 1, 30}] (* _Vaclav Kotesovec_, Sep 10 2019 *)
%o (PARI) /* Dirichlet convolution of mu(n) with Catalan numbers: */
%o {a(n) = sumdiv(n, d, moebius(n/d) * binomial(2*(d-1),d-1)/d)}
%o for(n=1,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 04 2015
%o (PARI) /* G.f. satisfies: Sum_{n>=1} A((x-x^2)^n) = x: */
%o {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]}
%o for(n=1,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 04 2015
%K nonn,easy
%O 1,4
%A _Erich Friedman_
|
