A098977 Triangle read by rows: counts ordered trees by number of edges and position of first edge that terminates at a vertex of outdegree 1.

1, 1, 1, 2, 2, 1, 4, 5, 3, 2, 9, 14, 9, 6, 4, 21, 42, 28, 19, 13, 9, 51, 132, 90, 62, 43, 30, 21, 127, 429, 297, 207, 145, 102, 72, 51, 323, 1430, 1001, 704, 497, 352, 250, 178, 127, 835, 4862, 3432, 2431, 1727, 1230, 878, 628, 450, 323, 2188, 16796, 11934, 8502

Offset: 1

Author

David Callan, Oct 24 2004

Keywords

• nonn
• tabl

Comments

T(n,k) = number of ordered trees on n edges whose k-th edge (in preorder or "walk around from root" order) is the first one that terminates at a vertex of outdegree 1 (k=0 if there is no such edge). The first column and the main diagonal (after initial entry) are Motzkin numbers (A001006). Each interior entry is the sum of its North and East neighbors.

Examples

			Table begins
\ k 0, 1, 2, ...
n
1 | 1
2 | 1, 1
3 | 2, 2, 1
4 | 4, 5, 3, 2
5 | 9, 14, 9, 6, 4
6 | 21, 42, 28, 19, 13, 9
7 | 51, 132, 90, 62, 43, 30, 21
8 |127, 429, 297, 207, 145, 102, 72, 51
T(4,2)=3 counts the following ordered trees (drawn down from root).
..|..../\..../|\..
./.\....|.....|...
.|......|.........
		

Crossrefs

Column k=1 is A000108 (apart from first term), k=2 is A000245, k=3 is A026012.

Programs

Mathematica

Clear[v] MotzkinNumber[n_]/;IntegerQ[n] && n>=0 := If[0<=n<=1, 1, Module[{x = 1, y = 1}, Do[temp = ((2*i + 1)*y + 3*(i - 1)*x)/(i + 2); x = y; y = temp, {i, 2, n}]; y]]; v[n_, 0]/; n>=1 := MotzkinNumber[n-1]; v[n_, k_]/; k>=n := 0; v[n_, k_]/; n>=2 && k==n-1 := MotzkinNumber[n-2]; v[n_, k_]/; n>=3 && 1<=k<=n-2 := v[n, k] = v[n, k+1]+v[n-1, k]; TableForm[Table[v[n, k], {n, 10}, {k, 0, n-1}]]

Formula

G.f. for column k=0 is (1 - z - (1-2*z-3*z^2)^(1/2))/(2*z^2) = Sum_{n>=1}T(n, 0)z^n. G.f. for columns k>=1 is (t*(1 - (1 - 4*z)^(1/2) - 2*z))/ (1 - t + t*(1 - 4*z)^(1/2) + t*z + (1 - 2*t*z - 3*t^2*z^2)^(1/2)) = Sum_{n>=2, 1<=k<=n-1}T(n, k)z^n*t^k.