|
Search: id:A069269
|
|
|
| A069269 |
|
Second level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 3rd level is A069270). |
|
+0
3
|
|
| 1, 1, 1, 1, 2, 3, 1, 3, 7, 12, 1, 4, 12, 30, 55, 1, 5, 18, 55, 143, 273, 1, 6, 25, 88, 273, 728, 1428, 1, 7, 33, 130, 455, 1428, 3876, 7752, 1, 8, 42, 182, 700, 2448, 7752, 21318, 43263, 1, 9, 52, 245, 1020, 3876, 13566, 43263, 120175, 246675, 1, 10, 63, 320, 1428
(list; table; graph; listen)
|
|
|
OFFSET
|
0,5
|
|
|
COMMENT
|
For the m-th level generalization of Catalan triangle T(n,k)=C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n>=k+m: T(n,k)=T(n-m+1,k+1)-T(n-m,k+1); and T(n,n)=T(n+m-1,n-1)=C((m+1)n,n)/(mn+1).
Reflected version of A110616 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 15 2007
With offset 1 for n and k, T(n,k) is (conjecturally) the number of permutations of [n] that avoid the patterns 4-2-3-1 and 4-2-5-1-3 and for which the last ascent ends at position k (k=1 if there are no ascents). For example, T(4,1) = 1 counts 4321; T(4,2) = 3 counts 1432, 2431, 3421; T(4,3) = 7 counts 1243, 1342, 2143, 2341, 3142, 3241, 4132. - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008
|
|
REFERENCES
|
M. H. Albert et al., Restricted permutations and queue jumping, Discrete Mathematics, Vol. 287, Issues 1-3, 2004, 129-133.
|
|
FORMULA
|
T(n, k)=C(n+2k, k)*(n-k+1)/(n+k+1). For n>=k+2: T(n, k)=T(n-1, k+1)-T(n-2, k+1). T(n, n)=T(n+1, n-1)=C(3n, n)/(2n+1).
|
|
EXAMPLE
|
Rows start 1; 1,1; 1,2,3; 1,3,7,12; 1,4,12,30,55; etc.
|
|
CROSSREFS
|
Columns include A000012, A000027, A055998. Righthand columns include (among others) A001764, A006013, A001764 (without first 1), A006629, A006630, A006630, A006631. Cf. triangles A007318, A009766, A069270.
Sequence in context: A062869 A102473 A011117 this_sequence A100324 A121424 A093768
Adjacent sequences: A069266 A069267 A069268 this_sequence A069270 A069271 A069272
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Henry Bottomley (se16(AT)btinternet.com), Mar 12 2002
|
|
|
Search completed in 0.002 seconds
|
