This site is supported by donations to The OEIS Foundation.

 Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 0
 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; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 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. LINKS 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. 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). ..|..../\..../|\.. ./.\....|.....|... .|......|......... 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}]] CROSSREFS Column k=1 is A000108 (apart from first term), k=2 is A000245, k=3 is A026012. Sequence in context: A064189 A273897 A063415 * A247311 A113547 A218580 Adjacent sequences:  A098974 A098975 A098976 * A098978 A098979 A098980 KEYWORD nonn,tabl AUTHOR David Callan, Oct 24 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 12 03:27 EST 2018. Contains 318052 sequences. (Running on oeis4.)