|
| |
|
|
A112413
|
|
Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and starting with exactly k UD's, where U=(1,1), D=(1,-1) (0<=k<=n).
|
|
0
| |
|
|
1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 9, 3, 1, 0, 1, 28, 9, 3, 1, 0, 1, 90, 28, 9, 3, 1, 0, 1, 297, 90, 28, 9, 3, 1, 0, 1, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 3432, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 11934, 3432, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 41990, 11934, 3432, 1001, 297, 90, 28, 9, 3, 1, 0
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,7
|
|
|
COMMENTS
| All columns, except for initial terms, yield A000245. Row sums yield the Catalan numbers (A000108).
Riordan array ((1-x)*c(x),x), c(x) the g.f. of A000108; equal to A125177*A130595 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 08 2009]
|
|
|
FORMULA
| T(n, k)=c(n-k)-c(n-k-1), where c(n)=binomial(2n, n)/(n+1) is the n-th Catalan number. G.f.=(1-z)C/(1-tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
|
|
|
EXAMPLE
| T(5,2)=3 because we have UDUDUUDDUD, UDUDUUDUDD and UDUDUUUDDD, where U=(1,1), D=(1,-1).
Triangle begins:
1;
0,1;
1,0,1;
3,1,0,1;
9,3,1,0,1;
28,9,3,1,0,1;
|
|
|
MAPLE
| T:=proc(n, k) local c: c:=n->binomial(2*n, n)/(n+1): if k<n then c(n-k)-c(n-k-1) elif k=n then 1 else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
|
|
|
CROSSREFS
| Cf. A000108, A000245.
Sequence in context: A151511 A048993 A193357 * A122960 A091480 A034374
Adjacent sequences: A112410 A112411 A112412 * A112414 A112415 A112416
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 08 2005
|
| |
|
|
