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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).
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).
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
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 18 2005
STATUS
approved