|
Search: id:A058987
|
|
|
| A058987 |
|
Catalan(n) - Motzkin(n-1). |
|
+0
1
|
|
| 0, 1, 3, 10, 33, 111, 378, 1303, 4539, 15961, 56598, 202214, 727389, 2632605, 9581211, 35047098, 128791323, 475281921, 1760726808, 6545921136, 24415415001, 91340016081, 342658850427, 1288774386909, 4858753673655, 18358309669651
(list; graph; listen)
|
|
|
OFFSET
|
1,3
|
|
|
COMMENT
|
Number of Dyck paths with a "small Capital N" (a rise then a fall then a rise) - this follows from the exercise on p. 238 of Stanley stating that Motzkin numbers equal to the ballot number without (1,-1,1). Since Ballot numbers are Catalan numbers, the result follows from the well-known bijection with Dyck paths.
a(n+1)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=Catalan(k+1) for k=0,1,...,n. - Michael Somos, Oct 07 2003
|
|
REFERENCES
|
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; cf. p. 238.
|
|
PROGRAM
|
(PARI) a(n)=if(n<1, 0, subst(polinterpolate(vector(n, k, binomial(2*k, k)/(k+1))), x, n+1))
|
|
CROSSREFS
|
A000108(n) - A001006(n-1).
Sequence in context: A113299 A126931 A071722 this_sequence A001558 A111639 A006535
Adjacent sequences: A058984 A058985 A058986 this_sequence A058988 A058989 A058990
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
You Seng Peng (giawgwan(AT)single.url.com.tw), Jan 17 2001
|
|
|
Search completed in 0.002 seconds
|
