

A143362


Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k protected vertices (0<=k<=n1). A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.


2



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, 1, 343, 481, 325, 205, 46, 28, 1, 1, 1004, 1491, 1286, 631, 351, 61, 36, 1, 1, 2938, 4929, 4280, 2861, 1101, 562, 78, 45, 1, 1, 8791, 15840, 15662, 10025, 5676, 1783, 855, 97
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OFFSET

1,4


COMMENTS

Row sums are the Catalan numbers (A000108).
Sum(k*T(n,k),k>=0) = A014301(n).
T(n,0) = A143363(n)


LINKS

Table of n, a(n) for n=1..63.
GiSang Cheon and Louis W. Shapiro, Protected points in ordered trees, Appl. Math. Letters, 21, 2008, 516520.


FORMULA

G.f.: G1, where G=G(t,z) satisfies G = 1/(1zG) + z(t1)(G1)/(1+zzG).


EXAMPLE

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.
Triangle starts:
1;
1,1;
3,1,1;
6,6,1,1;
17,13,10,1,1;
43,50,22,15,1,1;


MAPLE

eq:=G1/(1z*G)z*(t1)*(G1)/(1+zz*G): G:=RootOf(eq, G): Gser:=simplify(series(G1, 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..n1) end do; # yields sequence in triangular form


CROSSREFS

Cf. A000108, A014301, A143363.
Sequence in context: A146769 A189610 A172427 * A182823 A210866 A245474
Adjacent sequences: A143359 A143360 A143361 * A143363 A143364 A143365


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Aug 20 2008


STATUS

approved



