|
| |
|
|
A100754
|
|
Triangle read by rows: T(n,k)=number of hill-free Dyck paths (i.e. no peaks at height 1) of semilength n and having k peaks.
|
|
6
| |
|
|
1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 29, 13, 1, 1, 19, 73, 73, 19, 1, 1, 26, 151, 266, 151, 26, 1, 1, 34, 276, 749, 749, 276, 34, 1, 1, 43, 463, 1781, 2762, 1781, 463, 43, 1, 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 2,5
|
|
|
COMMENTS
| Row n has n-1 terms. Row sums yield the Fine numbers (A000957).
|
|
|
REFERENCES
| E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
|
|
|
FORMULA
| T(n, k)=sum([j/(n-j)]binomial(n-j, k-j)binomial(n-j, k), j=0..min(k, n-k)) (n>=2). G.f.=tzr/(1-tzr), where r=r(t, z) is the Narayana function defined by r=z(1+r)(1+tr).
|
|
|
EXAMPLE
| T(4,2)=4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD and UUU*DU*DDD, where U=(1,1), D=(1,-1) and * indicates the peaks.
Triangle starts:
1;
1,1;
1,4,1;
1,8,8,1;
1,13,29,13,1;
|
|
|
MAPLE
| T:=(n, k)->sum((j/(n-j))*binomial(n-j, k-j)*binomial(n-j, k), j=0..min(k, n-k)): for n from 2 to 13 do seq(T(n, k), k=1..n-1) od; # yields the sequence in triangular form
|
|
|
CROSSREFS
| Cf. A000957.
Sequence in context: A141541 A177947 A132789 * A174035 A055107 A128137
Adjacent sequences: A100751 A100752 A100753 * A100755 A100756 A100757
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 14 2005
|
| |
|
|
