A127158 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k branches of length 1 starting from the root (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 3, 5, 5, 0, 1, 7, 18, 9, 7, 0, 1, 20, 52, 37, 13, 9, 0, 1, 59, 168, 113, 60, 17, 11, 0, 1, 184, 546, 388, 190, 87, 21, 13, 0, 1, 593, 1826, 1313, 688, 283, 118, 25, 15, 0, 1, 1964, 6211, 4545, 2408, 1076, 392, 153, 29, 17, 0, 1, 6642, 21459

Offset: 0

Author

Emeric Deutsch, Mar 01 2007

Keywords

• nonn
• tabl

Comments

Row sums are the Catalan numbers (A000108). T(n,0)=A030238(n-2) for n>=2. Sum(k*T(n,k),k=0..n)=A026012(n-1) for n>=1.

Examples

			Triangle starts:
1;
0,1;
1,0,1;
1,3,0,1;
3,5,5,0,1;
7,18,9,7,0,1;
		

Crossrefs

Cf. A000108, A030238, A026012.

Programs

Maple

C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-t*z*C+t*z^2*C-z^2*C): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

Formula

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