A102004 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k branches of even length (n>=0, 0<=k<=floor(n/2)).

1, 1, 1, 1, 3, 2, 6, 7, 1, 16, 20, 6, 40, 64, 26, 2, 109, 196, 108, 16, 297, 619, 414, 96, 4, 836, 1940, 1557, 484, 45, 2377, 6142, 5690, 2247, 331, 9, 6869, 19454, 20535, 9792, 2010, 126, 20042, 61893, 73123, 40997, 10820, 1116, 21, 59071, 197280, 258220

Offset: 0

Author

Emeric Deutsch, Dec 25 2004

Keywords

• nonn
• tabf

Comments

Row n has 1+floor(n/2) terms.

Row sums are the Catalan numbers (A000108).

T(2n,n) = A001006(n-1) for n>=1 (the Motzkin numbers).

T(2n+1,n) = A005717(n+1) for n>=0.

Examples

			T(3,0)=3 because we have: (i) tree with 3 edges hanging from the root, (ii) tree with one edge hanging from the root, at the end of which 2 edges are hanging and (iii) tree with a path of length 3 hanging from the root.
Triangle starts:
1;
1;
1,   1;
3,   2;
6,   7, 1;
16, 20, 6;
		

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, A005717, A102003.

Programs

Maple

G:=1/2/(t*z^2+z)*(-z^2+z+1+t*z^2-sqrt(-5*z^2-6*t*z^3-2*z+2*z^3-3*t^2*z^4-2*t*z^2+2*t*z^4+1+z^4)): Gserz:=simplify(series(G,z=0,16)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(expand(coeff(Gserz,z^n))) od:for n from 0 to 14 do seq(coeff(t*P[n], t^k),k=1..1+floor(n/2)) od;

Formula

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