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.
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)
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
KEYWORD
sign,easy
AUTHOR
Paul D. Hanna, Jun 03 2006
STATUS
approved