A114586 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at odd levels (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1.

1, 1, 1, 3, 2, 1, 6, 8, 3, 1, 15, 22, 15, 4, 1, 36, 68, 52, 24, 5, 1, 91, 198, 191, 100, 35, 6, 1, 232, 586, 651, 425, 170, 48, 7, 1, 603, 1718, 2203, 1656, 820, 266, 63, 8, 1, 1585, 5047, 7285, 6299, 3591, 1435, 392, 80, 9, 1, 4213, 14808, 23832, 23164, 15155, 6972, 2338

Offset: 2

Author

Emeric Deutsch, Dec 11 2005

Keywords

• nonn
• tabl

Comments

Row sums are the Fine numbers (A000957). Column 0 yield the Riordan numbers (A005043). Sum(k*T(n,k),k=0..n-2)=A114587(n).

Examples

			T(5,2)=3 because we have UU(UD)DU(UD)DD, UUDU(UD)(UD)DD and UU(UD)(UD)DUDD, where U=(1,1), D=(1,-1) (the peaks at odd levels are shown between parentheses).
Triangle begins:
1;
1,1;
3,2,1;
6,8,3,1;
15,22,15,4,1;
		

Crossrefs

Cf. A000957, A005043, A114587, A114588, A100754.

Programs

Maple

G:=(t*z+z+1-sqrt(z^2*t^2+2*z^2*t-2*z*t-3*z^2-2*z+1))/2/z/(1+t+z)-1: 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-1) od; # yields sequence in triangular form

Formula

G.f.=G-1, where G=G(t, z) satisfies z(1+t+z)G^2-(1+z+tz)G+1=0.