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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
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OFFSET
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1,4
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COMMENTS
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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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LINKS
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Table of n, a(n) for n=1..59.
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FORMULA
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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
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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
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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
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Column k=1 is A000108 (apart from first term), k=2 is A000245, k=3 is A026012.
Sequence in context: A273897 A330792 A063415 * A247311 A113547 A218580
Adjacent sequences: A098974 A098975 A098976 * A098978 A098979 A098980
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KEYWORD
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nonn,tabl
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AUTHOR
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David Callan, Oct 24 2004
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STATUS
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approved
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