0,3

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 (deutsch(AT)duke.poly.edu), 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 (deutsch(AT)duke.poly.edu), Dec 14 2007

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

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 (deutsch(AT)duke.poly.edu), Dec 14 2007

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).

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 (deutsch(AT)duke.poly.edu), Dec 14 2007

Cf. A000108, A135337.

Sequence in context: A135305 A114463 A135309 this_sequence A135329 A114508 A139023

Adjacent sequences: A135328 A135329 A135330 this_sequence A135332 A135333 A135334

nonn,tabf

njas, Dec 07 2007

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007