0,3

Each of the rows 0,1,2 has one term; row n (n>=1) has ceil(n/2) terms. Row sums are the Catalan numbers (A000108). Column 0 yields A135334. - 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.

T(n,k)=2(k+1)Sum((-1)^(j-k)*binom(j+1,k+1)*binom(2n-2j-1,n), j=k..floor((n-1)/2))/(n+1) (n>=1). G.f.=1+zC^2/[1+(1-t)z^2*C^2], 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

4 1

10 4

29 12 1

90 36 6

290 114 24 1

960 376 86 8

3246 1272 303 40 1

...

T(4,1)=4 because we have UDU(UDDU)D, U(UDDU)DUD, U(UDDU)UDD and UUD(UDDU)D (the UDDU's starting at level 1 are shown between parentheses).

T:=proc(n, k) options operator, arrow: (2*k+2)*(sum((-1)^(j-k)*binomial(j+1, k+1)*binomial(2*n-2*j-1, n), j=k..floor((1/2)*n-1/2)))/(n+1) end proc: 1; for n to 13 do seq(T(n, k), k=0..ceil((n-2)*1/2)) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007

G:=1+z*C^2/(1+(1-t)*z^2*C^2): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=simplify(series(G, z=0, 16)): for n from 0 to 13 do P[n]:=sort(coeff(Gser, z, n)) end do: 1; for n to 13 do seq(coeff(P[n], t, j), j=0..floor((n-1)*1/2)) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007

Cf. A000108, A135334.

Sequence in context: A114506 A114848 A135330 this_sequence A048941 A135333 A124503

Adjacent sequences: A135325 A135326 A135327 this_sequence A135329 A135330 A135331

nonn,tabf

N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2007

Edited and extended by Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007