A114486 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k UUDD's starting at level 0; here U=(1,1), D=(1,-1) (0<=k<=floor(n/2)).

1, 1, 1, 1, 3, 2, 10, 3, 1, 31, 8, 3, 98, 27, 6, 1, 321, 88, 16, 4, 1078, 287, 54, 10, 1, 3686, 960, 183, 28, 5, 12789, 3280, 616, 95, 15, 1, 44919, 11378, 2106, 332, 45, 6, 159407, 39953, 7323, 1152, 155, 21, 1, 570704, 141752, 25785, 4028, 556, 68, 7, 2058817

Offset: 0

Author

Emeric Deutsch, Nov 30 2005

Keywords

• nonn
• tabf

Comments

Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(k*T(n,k),k=0..floor(n/2))=A000108(n-1) (the Catalan numbers). Column 0 yields A114487.

Examples

			T(5,2)=3 because we have UUDDUUDDUD, UUDDUDUUDD and UDUUDDUUDD, where U=(1,1), D=(1,-1).
Triangle starts:
1;
1;
1,1;
3,2;
10,3,1;
31,8,3;
98,27,6,1; ...
		

Links

• A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Crossrefs

Cf. A000108, A114487.

Programs

Maple

C:=(1-sqrt(1-4*z))/2/z: eq:=G=1+z*(C-z+t*z)*G: G:=solve(eq,G): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 15 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 15 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/2)) od; # yields sequence in triangular form

Formula

G.f. G=G(t, z) satisfies G=1+z(C-z+tz)G, where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. G=2/[1+2z^2-2tz^2+sqrt(1-4z)].