A192480 a(n) = n + A000108(n-1) for n > 1; a(0)=0, a(1)=1.

0, 1, 3, 5, 9, 19, 48, 139, 437, 1439, 4872, 16807, 58798, 208025, 742914, 2674455, 9694861, 35357687, 129644808, 477638719, 1767263210, 6564120441, 24466267042, 91482563663, 343059613674, 1289904147349, 4861946401478, 18367353072179, 69533550916032

Offset: 0

Author

Volkan Yildiz, Jul 01 2011

Keywords

• nonn

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.

Crossrefs

Cf. A000108.

Programs

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 *)

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

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