A114576 Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k UH's, where U=(1,1), H=(1,0) (0<=k<=floor(n/3)).

1, 1, 2, 3, 1, 6, 3, 11, 10, 23, 26, 2, 47, 70, 10, 102, 176, 45, 221, 449, 160, 5, 493, 1121, 539, 35, 1105, 2817, 1680, 196, 2516, 7031, 5082, 868, 14, 5763, 17604, 14856, 3486, 126, 13328, 43996, 42660, 12810, 840, 30995, 110147, 120338, 44640, 4410, 42

Offset: 0

Author

Emeric Deutsch, Dec 09 2005

Keywords

• nonn
• tabf

Comments

Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 0 yields A090344. Sum(k*T(n,k),k=0..floor(n/3))=A014531(n-2).

Examples

			T(4,1)=3 because we have H(UH)D, (UH)DH and (UH)HD, where U=(1,1), H=(1,0), D=(1,-1) (the UH's are shown between parentheses).
Triangle begins:
1;
1;
2;
3,1;
6,3;
11,10;
23,26,2;
47,70,10;
		

Crossrefs

Cf. A001006, A090344, A014531.

Programs

Maple

G:=(1-z-sqrt(1-2*z-3*z^2-4*z^3*t+4*z^3))/2/z^2/(1-z+t*z): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 16 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form

Formula

G.f.=[1-z-sqrt(1-2z-3z^2-4tz^3+4z^3)]/[2(1-z+tz)z^2].