0,3

Each of the rows 0,1, and 2 contains one entry. Row n contains n-1 entries (n>=2). Row sums are the Catalan numbers (A000108). Column 0 yields A135339. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007

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

G.f.=1+zC+z^2*C^3/[1+(1-t)zC], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). T(n,k)=d(0,k)*c(n-1)+Sum[(-1)^(j-k)*(j+3)binomial(j,k)binomial(2n-j-2,n),j=k..n-2]/(n+1), where c(m)=binomuial(2m,m)/(m+1)=A000108(m) is a Catalan number and d(0,k) is the Kronecker symbol. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007

Triangle begins:

1

1

2

4 1

11 2 1

32 7 2 1

99 22 8 2 1

318 73 26 9 2 1

1051 246 90 30 10 2 1

...

T(4,1)=2 because we have U(DUDU)UDD and UUD(DUDU)D; T(4,2)=1 because we have U(DU[DU)DU]D (the DUDU's starting at level 1 are shown between parentheses).

G:=1+z*C+z^2*C^3/(1+(1-t)*z*C): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=simplify(series(G, z=0, 17)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 12 do seq(coeff(P[n], t, j), j=0..n-2) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007

Cf. A000108, A135339.

Adjacent sequences: A135330 A135331 A135332 this_sequence A135334 A135335 A135336

Sequence in context: A135330 A135328 A048941 this_sequence A124503 A114499 A030730

nonn,tabf

njas, Dec 07 2007

Edited and extended by Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007