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A098977
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Triangle read by rows: counts ordered trees by number of edges and position of first edge that terminates at a vertex of outdegree 1.
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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.
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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.
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EXAMPLE
| 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).
..|..../\..../|\..
./.\....|.....|...
.|......|.........
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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}]]
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CROSSREFS
| Column k=1 is A000108 (apart from first term), k=2 is A000245, k=3 is A026012.
Sequence in context: A183191 A064189 A063415 * A113547 A115313 A048942
Adjacent sequences: A098974 A098975 A098976 * A098978 A098979 A098980
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KEYWORD
| nonn
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AUTHOR
| David Callan (callan(AT)stat.wisc.edu), Oct 24 2004
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