A100400 Triangle read by rows: T(n,k) is the number of nonroot nodes of outdegree k (0<=k<=n-1) in all non-crossing trees with n edges.

1, 4, 2, 21, 12, 3, 120, 72, 24, 4, 715, 440, 165, 40, 5, 4368, 2730, 1092, 312, 60, 6, 27132, 17136, 7140, 2240, 525, 84, 7, 170544, 108528, 46512, 15504, 4080, 816, 112, 8, 1081575, 692208, 302841, 105336, 29925, 6840, 1197, 144, 9, 6906900, 4440150, 1973400, 708400, 212520, 53130, 10780, 1680, 180, 10

Offset: 1

Author

Emeric Deutsch, Dec 30 2004

Keywords

• nonn
• tabl

Comments

Row n contains n terms. Row sums yield A004319. Column 0 yields A045721.

Examples

			T(2,1)=2 because in the non-crossing trees /_, _\ and /\ we have 2 nonroot nodes of outdegree 1.
Triangle begins:
1;
4,2;
21,12,3;
120,72,24,4;
		

Links

• E. Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87.

Crossrefs

Cf. A001764, A004319, A045721.

Programs

Maple

for n from 1 to 10 do seq((k+1)*binomial(3*n-k-2,2*n-1),k=0..n-1) od; # yields sequence in triangular form

Formula

T(n, k) = (k+1)binomial(3n-k-2, 2n-1) (0<=k<=n-1).