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A143362 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k protected vertices (0<=k<=n-1). A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

%I A143362 #6 Jun 26 2015 11:10:50
%S A143362 1,1,1,3,1,1,6,6,1,1,17,13,10,1,1,43,50,22,15,1,1,123,141,109,33,21,1,
%T A143362 1,343,481,325,205,46,28,1,1,1004,1491,1286,631,351,61,36,1,1,2938,
%U A143362 4929,4280,2861,1101,562,78,45,1,1,8791,15840,15662,10025,5676,1783,855,97
%N A143362 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k protected vertices (0<=k<=n-1). A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.
%C A143362 Row sums are the Catalan numbers (A000108).
%C A143362 Sum(k*T(n,k),k>=0) = A014301(n).
%C A143362 T(n,0) = A143363(n)
%H A143362 Gi-Sang Cheon and Louis W. Shapiro, <a href="http://dx.doi.org/10.1016/j.aml.2007.07.001">Protected points in ordered trees</a>, Appl. Math. Letters, 21, 2008, 516-520.
%F A143362 G.f.: G-1, where G=G(t,z) satisfies G = 1/(1-zG) + z(t-1)(G-1)/(1+z-zG).
%e A143362 T(3,2)=1 because among the five ordered trees with 3 edges only the path tree has 2 vertices at least two edges away from the leaf.
%e A143362 Triangle starts:
%e A143362 1;
%e A143362 1,1;
%e A143362 3,1,1;
%e A143362 6,6,1,1;
%e A143362 17,13,10,1,1;
%e A143362 43,50,22,15,1,1;
%p A143362 eq:=G-1/(1-z*G)-z*(t-1)*(G-1)/(1+z-z*G): G:=RootOf(eq,G): Gser:=simplify(series(G-1,z=0,13)): for n to 11 do P[n]:=sort(expand(coeff(Gser,z,n))) end do: for n to 11 do seq(coeff(P[n],t,j),j=0..n-1) end do; # yields sequence in triangular form
%Y A143362 Cf. A000108, A014301, A143363.
%K A143362 nonn,tabl
%O A143362 1,4
%A A143362 _Emeric Deutsch_, Aug 20 2008