

A114626


Triangle read by rows: T(n,k) is the number of hillfree Dyck paths of semilength n and having k peaks at level 2; 0<= k<=n1, n>=2 (a Dyck path is said to be hillfree if it has no peaks at level 1).


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
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OFFSET

2,6


COMMENTS

Row n has n terms (n>=2). Row sums yield the Fine numbers (A000957). T(n,0)=A114627(n3). Sum(kT(n,k),k=0..n1)=A114495(n).


LINKS



FORMULA

G.f.=(1+ztzzC)/[1+z+z^2tztz^2z(1+z)C], where C=[1sqrt(14z)]/(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:=(1sqrt(14*z))/2/z: G:=(1+zt*zz*C)/(1+z+z^2t*zt*z^2z*(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



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



