A118972 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having length of first descent equal to k (1<=k<=n; n>=1). A hill in a Dyck path is a peak at level 1.
0, 0, 1, 1, 0, 1, 3, 2, 0, 1, 10, 5, 2, 0, 1, 33, 16, 5, 2, 0, 1, 111, 51, 16, 5, 2, 0, 1, 379, 168, 51, 16, 5, 2, 0, 1, 1312, 565, 168, 51, 16, 5, 2, 0, 1, 4596, 1934, 565, 168, 51, 16, 5, 2, 0, 1, 16266, 6716, 1934, 565, 168, 51, 16, 5, 2, 0, 1, 58082, 23604, 6716, 1934, 565, 168
Offset: 1
Examples
T(5,2)=5 because we have uu(dd)uududd, uu(dd)uuuddd,uuu(dd)uuddd,uuu(dd)ududd and uuuu(dd)uddd, where u=(1,1), d=(1,-1) (the first descents are shown between parentheses). Triangle starts: 0; 0,1; 1,0,1; 3,2,0,1; 10,5,2,0,1; 33,16,5,2,0,1; ...
Links
- Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Programs
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Maple
F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: G:=t*z^2*C*F*(C-(1-t)/(1-t*z)): Gser:=simplify(series(G,z=0,15)): for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 1 to 12 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
Formula
G.f.: t*z^2*C*F*(C-(1-t)/(1-t*z)), where F = (1-sqrt(1-4*z))/(z*(3-sqrt(1-4*z))) and C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
Comments