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

Author

Emeric Deutsch, Dec 18 2005

Keywords

• nonn
• tabf

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).

Examples

			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;
		

Crossrefs

Cf. A000957, A114627, A114495.

Programs

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

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.