|
| |
|
|
A101371
|
|
Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges and k leaves at level 1.
|
|
0
| |
|
|
1, 0, 1, 2, 0, 1, 7, 4, 0, 1, 34, 14, 6, 0, 1, 171, 72, 21, 8, 0, 1, 905, 370, 114, 28, 10, 0, 1, 4952, 1995, 597, 160, 35, 12, 0, 1, 27802, 11064, 3278, 852, 210, 42, 14, 0, 1, 159254, 62774, 18420, 4762, 1135, 264, 49, 16, 0, 1, 927081, 362614, 105618, 27104, 6455, 1446, 322, 56, 18, 0, 1
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| Row n has n+1 terms. Row sums give A001764. Column 0 gives A023053.
|
|
|
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)=sum((-1)^i*[(k+i+1)/(2n-k+1)]binomial(k+i, i)binomial(3n-2k-2i, n-k-i), i=0..n-k) (0<=k<=n). G.f.=g/(1+zg-tzg), where g=1+zg^3.
|
|
|
EXAMPLE
| Triangle begins:
1;
0,1;
2,0,1;
7,4,0,1;
34,14,6,0,1;
171,72,21,8,0,1;
|
|
|
MAPLE
| T:=proc(n, k) if k<=n then sum((-1)^i*(k+i+1)*binomial(k+i, i)*binomial(3*n-2*k-2*i, n-k-i)/(2*n-k-i+1), i=0..n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
|
|
|
CROSSREFS
| Cf. A001764, A023053.
Sequence in context: A138106 A131689 A114329 * A154974 A078341 A199459
Adjacent sequences: A101368 A101369 A101370 * A101372 A101373 A101374
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 14 2005
|
| |
|
|
