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.

1, 1, 1, 0, -6, -33, -143, -572, -2210, -8398, -31654, -118864, -445740, -1671525, -6273135, -23571780, -88704330, -334347090, -1262330850, -4773905760, -18083762580, -68611922730, -260725306374, -992233959480, -3781513867796, -14431491699548, -55147299002348

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..27.

Formula

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).

a(n) = C(2*n,n)/(n+1) - 4*C(2*n-1,n-2)/(n+2).

a(n) = 3*Catalan(n) - Catalan(n+1). - David Callan, Nov 21 2006

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

From Peter Bala, Feb 02 2024: (Start)

a(n) = 3*(-1)^n*Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*(2*k + 2)!/((k + 3)!*k!).

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.g.f.: exp(2*x)*(2*BesselI(0, 2*x) - 3*BesselI(1, 2*x) + BesselI(2, 2*x)) - 2. - Stefano Spezia, Dec 31 2024

Example

A(x) = x + x^2 + x^3 - 6*x^5 - 33*x^6 - 143*x^7 - 572*x^8 - 2210*x^9 + ...
A(x) = x*C(x)^2 - x^2*C(x)^4 where C(x) is Catalan function so that:
x*C(x)^2 = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ...
x^2*C(x)^4 = x^2 + 4*x^3 + 14*x^4 + 48*x^5 + 165*x^6 + 572*x^7 + ...

Mathematica

f[n_]:=3*CatalanNumber[n] -CatalanNumber[n+1]; Array[f, 30, 1] (* Vincenzo Librandi, Jan 02 2025 *)

Prog

(PARI) a(n)=binomial(2*n, n)/(n+1)-4*binomial(2*n-1, n-2)/(n+2)
(Magma) [3*Catalan(n) - Catalan(n+1): n in [1..30]]; // Vincenzo Librandi, Jan 02 2025

Crossrefs

Cf. A120010 (composition transpose), A000108 (Catalan), A000245.

Cf. A003517 (|a(n+1)|-|a(n)|). - Olivier Gérard, Oct 11 2012

Sequence in context: A263479 A073375 A089097 * A074087 A297592 A255613

Adjacent sequences: A120006 A120007 A120008 * A120010 A120011 A120012

Keyword

sign,easy,changed

Author

Paul D. Hanna, Jun 03 2006

Status

approved