A119011 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k valleys strictly above the x-axis (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1.
1, 1, 1, 2, 3, 1, 3, 8, 6, 1, 5, 18, 23, 10, 1, 8, 38, 70, 54, 15, 1, 13, 76, 186, 215, 110, 21, 1, 21, 147, 451, 710, 560, 202, 28, 1, 34, 277, 1025, 2065, 2269, 1288, 343, 36, 1, 55, 512, 2220, 5480, 7854, 6321, 2688, 548, 45, 1, 89, 932, 4634, 13574, 24227, 25830
Offset: 2
Examples
T(5,2)=6 because we have uud|ud|uuddd, uuudd|ud|udd, uud|uudd|udd, uuud|ud|uddd, uuud|udd|udd and uud|uud|uddd (the valleys above the x-axis are marked with |). Triangle starts: 1; 1,1; 2,3,1; 3,8,6,1; 5,18,23,10,1;
Links
- E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Programs
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Maple
G:=2*t/(2*t+z*t+z-1+sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))-1: Gser:=simplify(series(G,z=0,15)): for n from 2 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 2 to 12 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form
Formula
G.f.: G(t,z)=1/[1-zr(t,z)]-1, where r=r(t,z) is the Narayana function, defined by (1+r)(1+tr)z=r, r(t,0)=0. See Maple program for the explicit form of G(t,z).
Comments