Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k ascents of length 4 (0<=k<=floor(n/4)). Also number of ordered trees with n edges which have k vertices of outdegree 4.
T(5,1)=5 because we have UDUUUUDDDD, UUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD and UUUUDUDDDD, where U=(1,1), D=(1,-1).
Triangle starts:
1;
1;
2;
5;
13,1;
37,5;
111,21;
345,84;
1104,322,4;
3611,1215,36;
MAPLE
Order:=20: Y:=solve(series((Y-Y^2)/(1-(1-t)*Y^4+(1-t)*Y^5), Y)=z, Y): 1; for n from 1 to 17 do seq(coeff(t*coeff(Y, z^(n+1)), t^j), j=1..1+floor(n/4)) od; # yields sequence in triangular form
