|
Search: id:A091320
|
|
|
| A091320 |
|
Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges and k leaves. |
|
+0
1
|
|
| 1, 2, 1, 4, 7, 1, 8, 30, 16, 1, 16, 104, 122, 30, 1, 32, 320, 660, 365, 50, 1, 64, 912, 2920, 2875, 903, 77, 1, 128, 2464, 11312, 17430, 9856, 1960, 112, 1, 256, 6400, 39872, 88592, 78974, 28560, 3864, 156, 1, 512, 16128, 130944, 396480, 512316, 294042
(list; table; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
T(n,k) is the number of even trees with 2n edges and k-1 jumps. An even tree is an ordered tree in which each vertex has an even outdegree. In the preorder traversal of an ordered tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 19 2007
|
|
REFERENCES
|
P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discrete Math. 204 (1999), 203-229.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
|
|
FORMULA
|
T(n, k)=(1/n)*binomial(n, k)*sum(2^(n+1-2k+j)*binomial(n, j)*binomial(n-k, k-1-j), j=0..n). G.f. G(t, z) satisfies zG^3 - (1 + z - tz)G + 1 = 0.
|
|
EXAMPLE
|
1; 2,1; 4,7,1; 8,30,16,1; 16,104,122,30,1;
|
|
MAPLE
|
T := proc(n, k) if k=n then 1 else (1/n)*binomial(n, k)*sum(2^(n+1-2*k+j)*binomial(n, j)*binomial(n-k, k-1-j), j=0..n) fi end: seq(seq(T(n, k), k=1..n), n=1..12);
|
|
CROSSREFS
|
Row sums give A001764.
Sequence in context: A071948 A121722 A059579 this_sequence A048787 A030102 A072010
Adjacent sequences: A091317 A091318 A091319 this_sequence A091321 A091322 A091323
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 24 2004
|
|
|
Search completed in 0.002 seconds
|
