1,4

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.

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.

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).

..|..../\..../|\..

./.\....|.....|...

.|......|.........

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}]]

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

Adjacent sequences: A098974 A098975 A098976 this_sequence A098978 A098979 A098980

Sequence in context: A105306 A064189 A063415 this_sequence A113547 A115313 A048942

nonn

David Callan (callan(AT)stat.wisc.edu), Oct 24 2004