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 A120009 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. 8

%I

%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*CatalanNumber[n] - CatalanNumber[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

%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).

%Y cf. A003517 (|a(n+1)|-|a(n)|). From _Olivier GĂ©rard_, Oct 11 2012

%K sign

%O 1,5

%A _Paul D. Hanna_, Jun 03 2006

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Last modified August 8 14:04 EDT 2020. Contains 336298 sequences. (Running on oeis4.)