|
%I #25 Feb 03 2024 10:17:14
%S 1,1,1,0,-6,-33,-143,-572,-2210,-8398,-31654,-118864,-445740,-1671525,
%T -6273135,-23571780,-88704330,-334347090,-1262330850,-4773905760,
%U -18083762580,-68611922730,-260725306374,-992233959480,-3781513867796,-14431491699548,-55147299002348
%N G.f.: A(x) = (x-x^2) o x/(1-x) o (1-sqrt(1-4*x))/2, a composition of functions involving the Catalan function and its inverse.
%C The n-th self-composition of A(x) is: (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2. See A120010 for the transpose of the composition of the same functions.
%F G.f.: A(x) = ((1-3*x)*sqrt(1-4*x) - (1-x)*(1-4*x))/(2*x^2) = x*C(x)^2 - x^2*C(x)^4 where C(x) is the Catalan function (A000108).
%F a(n) = C(2*n,n)/(n+1) - 4*C(2*n-1,n-2)/(n+2).
%F a(n) = 3*Catalan(n) - Catalan(n+1). - _David Callan_, Nov 21 2006
%F D-finite with recurrence: (n+2)*a(n) + (-7*n-2)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Jan 20 2020, corrected Feb 16 2020
%F From _Peter Bala_, Feb 02 2024: (Start)
%F a(n) = 3*(-1)^n*Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*(2*k + 2)!/((k + 3)!*k!).
%F G.f.: x/(1 - 4*x)*c(-x/(1 - 4*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%e A(x) = x + x^2 + x^3 - 6*x^5 - 33*x^6 - 143*x^7 - 572*x^8 - 2210*x^9 + ...
%e A(x) = x*C(x)^2 - x^2*C(x)^4 where C(x) is Catalan function so that:
%e x*C(x)^2 = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ...
%e x^2*C(x)^4 = x^2 + 4*x^3 + 14*x^4 + 48*x^5 + 165*x^6 + 572*x^7 + ...
%o (PARI) a(n)=binomial(2*n,n)/(n+1)-4*binomial(2*n-1,n-2)/(n+2)
%Y Cf. A120010 (composition transpose), A000108 (Catalan), A000245.
%Y cf. A003517 (|a(n+1)|-|a(n)|). - _Olivier Gérard_, Oct 11 2012
%K sign,easy
%O 1,5
%A _Paul D. Hanna_, Jun 03 2006
| 