A114626 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at level 2; 0<= k<=n-1, n>=2 (a Dyck path is said to be hill-free if it has no peaks at level 1).

0, 1, 1, 0, 1, 2, 2, 1, 1, 6, 6, 3, 2, 1, 19, 17, 12, 5, 3, 1, 61, 56, 36, 20, 8, 4, 1, 202, 185, 120, 66, 31, 12, 5, 1, 683, 624, 409, 224, 110, 46, 17, 6, 1, 2348, 2144, 1408, 784, 385, 172, 66, 23, 7, 1, 8184, 7468, 4920, 2760, 1380, 624, 257, 92, 30, 8, 1, 28855, 26317

Offset

2,6

Comments

Row n has n terms (n>=2). Row sums yield the Fine numbers (A000957). T(n,0)=A114627(n-3). Sum(kT(n,k),k=0..n-1)=A114495(n).

Links

Table of n, a(n) for n=2..68.

Formula

G.f.=(1+z-tz-zC)/[1+z+z^2-tz-tz^2-z(1+z)C], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

Example

T(5,2)=3 because we have U(UD)(UD)UUDDD, UUUDD(UD)(UD)D and U(UD)UUDD(UD)D, where U=(1,1), D=(1,-1) (the peaks at level 2 are shown between parentheses).
Triangle begins:
0,1;
1,0,1;
2,2,1,1;
6,6,3,2,1;
19,17,12,5,3,1;

Maple

C:=(1-sqrt(1-4*z))/2/z: G:=(1+z-t*z-z*C)/(1+z+z^2-t*z-t*z^2-z*(1+z)*C): Gser:=simplify(series(G, z=0, 15)): for n from 2 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 2 to 12 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form

Crossrefs

Cf. A000957, A114627, A114495.

Sequence in context: A370207 A086610 A141760 * A221916 A124773 A129177

Adjacent sequences: A114623 A114624 A114625 * A114627 A114628 A114629

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 18 2005

Status

approved