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

0,3

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

Links

Table of n, a(n) for n=0..40.

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

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

Example

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).

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

Crossrefs

Cf. A000108, A135337.

Sequence in context: A135305 A114463 A135309 * A135329 A114508 A243366

Adjacent sequences: A135328 A135329 A135330 * A135332 A135333 A135334

Keyword

nonn,tabf

Author

N. J. A. Sloane, Dec 07 2007

Extensions

More terms from Emeric Deutsch, Dec 14 2007

Status

approved