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, 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

Offset: 0

Author

Emeric Deutsch, Dec 23 2004

Keywords

• nonn
• tabl

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,n-2) = A025566(n) for n>=2.

Examples

			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;
		

Links

• Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees and branch lengths, Discrete Math., 282, 2004, 89-94.
• J. Riordan, Enumeration of plane trees by branches and endpoints, J. Comb. Theory (A) 19, 1975, 214-222.

Crossrefs

Cf. A000108, A001006, A025566, A102004.

Programs

Maple

G:=1/2/(z^2+t*z)*(t*z+1-sqrt(1-2*t*z-3*t^2*z^2-4*z^2-4*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

Formula

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