OFFSET
0,3
COMMENTS
a(n) is the number of components of the n-th Catalan tree A_n.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
V. Yildiz, Catalan Tree & Parity of some sequences which are related to Catalan numbers, arXiv:1106.5187 [math.CO], 2011.
FORMULA
G.f.: (2*x^2*(2-x)+(1-x)^2*(1-sqrt(1-4*x)))/(2*(1-x)^2).
For large n, a(n) ~ (2^(2n) +n^2*sqrt(Pi*n)) / sqrt(Pi *n^3).
Conjecture: n*(3*n^2-16*n+19)*a(n) +(-15*n^3+95*n^2-188*n+120)*a(n-1) +2*(2*n-5)*(3*n^2-10*n+6)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
MAPLE
C := proc(n) binomial(2*n, n)/(n+1) ; end proc:
A192480 := proc(n) if n <=1 then n; else n+C(n-1) ; end if; end proc:
seq(A192480(n), n=0..40) ; # R. J. Mathar, Jul 13 2011
MATHEMATICA
CoefficientList[Series[(2*x^2*(2 - x) + (1 - x)^2*(1 - Sqrt[1 - 4*x]))/(2*(1 - x)^2), {x, 0, 50}], x] (* G. C. Greubel, Mar 28 2017 *)
PROG
(PARI) x='x+O('x^50); concat([0], Vec((2*x^2*(2-x)+(1-x)^2*(1-sqrt(1-4*x)))/(2*(1-x)^2))) \\ G. C. Greubel, Mar 28 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Volkan Yildiz, Jul 01 2011
STATUS
approved
