A135331 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k DUUU's starting at level 1.

1, 1, 2, 5, 13, 1, 36, 6, 105, 27, 320, 108, 1, 1011, 409, 10, 3289, 1508, 65, 10957, 5491, 347, 1, 37216, 19898, 1658, 14, 128435, 72063, 7395, 119, 449142, 261436, 31527, 794, 1, 1588228, 951258, 130353, 4583, 18

Offset: 0

Author

N. J. A. Sloane, Dec 07 2007

Keywords

• nonn
• tabf

Comments

The formula for T(n,k)=l_{n,k} in the reference (p. 2919) does not appear to work (a typo is possible). - Emeric Deutsch, Dec 14 2007

Row 0 has 1 term; row n (n>=1) has floor((n+2)/3) terms. Row sums are the Catalan numbers (A000108). Column 0 yields A135337. - Emeric Deutsch, Dec 14 2007

Examples

			Triangle begins:
1
1
2
5
13 1
36 6
105 27
320 108 1
1011 409 10
3289 1508 65
10957 5491 347 1
...
T(5,1)=6 because we have U(DUUU)UDDDD, U(DUUU)DUDDD, U(DUUU)DDUDD, U(DUUU)DDDUD, UDU(DUUU)DDD and UUD(DUUU)DDD (the DUUU's starting at level 1 are shown between parentheses).
		

Links

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

Crossrefs

Cf. A000108, A135337.

Programs

Maple

G:=1+z*C^2/(1+(1-t)*z^3*C^4): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=simplify(series(G,z=0,16)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) end do: 1; for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((n-1)*1/3)) end do; # yields sequence in triangular form; Emeric Deutsch, Dec 14 2007

Formula

G.f.: G(t,z)=1+zC^2/[1+(1-t)z^3*C^4], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch, Dec 14 2007

Extensions

More terms from Emeric Deutsch, Dec 14 2007