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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
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 (list; graph; refs; listen; history; text; internal format)
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*CatalanNumber[n] - CatalanNumber[n+1]. - David Callan, Nov 21 2006

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 +...

PROG

(PARI) a(n)=binomial(2*n, n)/(n+1)-4*binomial(2*n-1, n-2)/(n+2)

CROSSREFS

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

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

Sequence in context: A263479 A073375 A089097 * A074087 A297592 A255613

Adjacent sequences:  A120006 A120007 A120008 * A120010 A120011 A120012

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jun 03 2006

STATUS

approved

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Last modified March 23 11:10 EDT 2019. Contains 321424 sequences. (Running on oeis4.)