A114508 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.

1, 1, 2, 5, 13, 1, 37, 5, 111, 21, 345, 84, 1104, 322, 4, 3611, 1215, 36, 12016, 4555, 225, 40548, 17028, 1210, 138414, 63636, 5940, 22, 477076, 238004, 27534, 286, 1657956, 891268, 122850, 2366, 5802920, 3342375, 533625, 15925, 20436910, 12552580

Offset: 0

Author

Emeric Deutsch, Dec 03 2005

Keywords

• nonn
• tabf

Comments

Row n has 1+floor(n/4) terms. Row sums yield the Catalan numbers (A000108). Column 0 yields A114509. Sum(kT(n,k),k=0..floor(n/4))=binomial(2n-5,n-4) (A002054).

Examples

			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;
		

Crossrefs

Cf. A102402, A114506, A000108, A002054, A114509.

Programs

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

Formula

G.f. G=G(t, z) satisfies (1-t)z^5*G^5-(1-t)z^4*G^4+zG^2-G+1=0.