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

2,4

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).

Links

Table of n, a(n) for n=2..63.

Formula

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

Example

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;

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

Crossrefs

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

Sequence in context: A257558 A202390 A210858 * A052174 A227790 A181897

Adjacent sequences: A114583 A114584 A114585 * A114587 A114588 A114589

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 11 2005

Status

approved