A127541 Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k even-length branches starting at the root (0<=k<=n).

1, 1, 1, 1, 3, 2, 8, 5, 1, 24, 15, 3, 75, 46, 10, 1, 243, 148, 34, 4, 808, 489, 116, 16, 1, 2742, 1652, 402, 61, 5, 9458, 5678, 1408, 228, 23, 1, 33062, 19792, 4982, 847, 97, 6, 116868, 69798, 17783, 3138, 393, 31, 1, 417022, 248577, 63967, 11627, 1557, 143, 7

Offset: 0

Author

Emeric Deutsch, Mar 01 2007

Keywords

• nonn
• tabf

Comments

Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). T(n,0)=A000958(n-1). Sum(k*T(n,k),k=0..floor(n/2))=A127540(n-1).

Examples

			T(2,0)=1 because we have the tree /\.
Triangle starts:
1;
1;
1,1;
3,2;
8,5,1;
24,15,3;
		

Crossrefs

Cf. A000108, A000958, A127538, A127540.

Programs

Maple

C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)/(1+z-z*C-t*z^2*C): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

Formula

G.f.=(1+z)/(1+z-z*C-tz^2*C), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.