A129170 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n such that the sum of the height of the peaks is k (n >= 0; n <= k <= floor((n+1)^2/4)).
1, 0, 1, 0, 0, 3, 0, 0, 0, 8, 2, 0, 0, 0, 0, 21, 9, 6, 0, 0, 0, 0, 0, 55, 33, 29, 16, 4, 0, 0, 0, 0, 0, 0, 144, 111, 114, 84, 60, 18, 12, 0, 0, 0, 0, 0, 0, 0, 377, 355, 409, 356, 305, 199, 120, 58, 32, 8, 0, 0, 0, 0, 0, 0, 0, 0, 987, 1098, 1389, 1365, 1308, 1032, 843, 507, 372, 204, 120
Offset: 0
Examples
T(3,4)=2 because we have UUDUDD and UUDUDL. Triangle starts: 1; 0, 1; 0, 0, 3; 0, 0, 0, 8, 2; 0, 0, 0, 0, 21, 9, 6; 0, 0, 0, 0, 0, 55, 33, 29, 16, 4;
Links
- E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
Programs
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Maple
H:=(1+z*h[1]-z)/(1-z*h[1]+z-z*t*x): for n from 1 to 11 do h[n]:=(1+z*h[n+1]-z)/(1-z*h[n+1]+z-z*t^(n+1)*x) od: h[12]:=0: x:=1: G:=simplify(H): Gser:=simplify(series(G,z=0,11)): for n from 0 to 9 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 9 do seq(coeff(P[n],t,j),j=0..floor((n+1)^2/4)) od; # yields sequence in triangular form
Formula
Sum_{k>=0} k*T(n,k) = A129171(n).
G.f.: G(t,z)=H(t,1,z), where H(t,x,z)=1+z[H(t,tx,z)-1+tx]H(t,x,z)+z[H(t,tx,z)-1] (H(t,x,z) is the trivariate g.f. for skew Dyck paths according to sum of the height of the peaks, number of peaks and semilength, marked by t,x and z, respectively).
Comments