

A102003


Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k branches of odd length (n>=0, 0<=k<=n).


1



1, 0, 1, 1, 0, 1, 0, 3, 0, 2, 2, 0, 8, 0, 4, 0, 11, 0, 22, 0, 9, 5, 0, 45, 0, 61, 0, 21, 0, 41, 0, 166, 0, 171, 0, 51, 14, 0, 226, 0, 580, 0, 483, 0, 127, 0, 154, 0, 1050, 0, 1962, 0, 1373, 0, 323, 42, 0, 1070, 0, 4430, 0, 6496, 0, 3923, 0, 835, 0, 582, 0, 6005, 0, 17570, 0, 21184, 0, 11257, 0, 2188
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OFFSET

0,8


COMMENTS

Row n has n+1 terms.
Column 0 yields the Catalan numbers (A000108) alternating with 0's.
The diagonal entries are the Motzkin numbers (A001006).
T(n,n2) = A025566(n) for n>=2.


LINKS

Table of n, a(n) for n=0..77.
Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees and branch lengths, Discrete Math., 282, 2004, 8994.
J. Riordan, Enumeration of plane trees by branches and endpoints, J. Comb. Theory (A) 19, 1975, 214222.


FORMULA

G.f. G = G(t,z) satisfies z(t+z)G^2(1+tz)G+1+tz=0.


EXAMPLE

T(3,3)=2 because we have (i) a tree with 3 edges hanging from the root and (ii) a tree with one edge hanging from the root, at the end of which 2 edges are hanging.
Triangle starts:
1;
0,1;
1,0,1;
0,3,0,2;
2,0,8,0,4;


MAPLE

G:=1/2/(z^2+t*z)*(t*z+1sqrt(12*t*z3*t^2*z^24*z^24*t*z^3)): Gserz:=simplify(series(G, z=0, 14)):P[0]:=1: for n from 1 to 12 do P[n]:=sort(expand(coeff(Gserz, z^n))) od: for n from 0 to 12 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields the sequence in triangular form


CROSSREFS

Cf. A000108, A001006, A025566, A102004.
Sequence in context: A127913 A135991 A279631 * A176314 A004587 A104609
Adjacent sequences: A102000 A102001 A102002 * A102004 A102005 A102006


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Dec 23 2004


STATUS

approved



