|
%I #28 Mar 29 2017 20:26:53
%S 0,1,3,5,9,19,48,139,437,1439,4872,16807,58798,208025,742914,2674455,
%T 9694861,35357687,129644808,477638719,1767263210,6564120441,
%U 24466267042,91482563663,343059613674,1289904147349,4861946401478,18367353072179,69533550916032
%N a(n) = n + A000108(n-1) for n > 1; a(0)=0, a(1)=1.
%C a(n) is the number of components of the n-th Catalan tree A_n.
%H G. C. Greubel, <a href="/A192480/b192480.txt">Table of n, a(n) for n = 0..1000</a>
%H V. Yildiz, <a href="http://arxiv.org/abs/1106.5187">Catalan Tree & Parity of some sequences which are related to Catalan numbers</a>, arXiv:1106.5187 [math.CO], 2011.
%F G.f.: (2*x^2*(2-x)+(1-x)^2*(1-sqrt(1-4*x)))/(2*(1-x)^2).
%F For large n, a(n) ~ (2^(2n) +n^2*sqrt(Pi*n)) / sqrt(Pi *n^3).
%F 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
%p C := proc(n) binomial(2*n,n)/(n+1) ; end proc:
%p A192480 := proc(n) if n <=1 then n; else n+C(n-1) ; end if; end proc:
%p seq(A192480(n),n=0..40) ; # _R. J. Mathar_, Jul 13 2011
%t 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 *)
%o (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
%Y Cf. A000108.
%K nonn
%O 0,3
%A _Volkan Yildiz_, Jul 01 2011
|
