0,3

Each of 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, Dec 13 2007

Alois P. Heinz, Rows n = 0..150, flattened

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) = binomial(2m,m)/(m+1) = A000108(m) is a Catalan number and d(0,k) is the Kronecker symbol. - Emeric Deutsch, 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, Dec 13 2007

# Alternative:

b:= proc(x, y, t) option remember; expand(`if`(x=0, 1,

`if`(y<x, b(x-1, y+1, [1, 3, 1, 3][t])*`if`(t=4, z, 1), 0)+

`if`(y>0, b(x-1, y-1, `if`(y=1, [2, 1, 4, 1][t], 1)), 0)))

end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):

seq(T(n), n=0..14); # Alois P. Heinz, Sep 28 2015

b[x_, y_, t_] := b[x, y, t] = Expand[If[x == 0, 1, If[y < x, b[x - 1, y + 1, {1, 3, 1, 3}[[t]]]*If[t == 4, z, 1], 0] + If[y > 0, b[x - 1, y - 1, If[y == 1, {2, 1, 4, 1}[[t]], 1]], 0]]]; T[n_] := Function [p, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 14 2016, after Alois P. Heinz *)

Cf. A000108, A135339.

Sequence in context: A048941 A308300 A246188 * A124503 A114499 A030730

Adjacent sequences: A135330 A135331 A135332 * A135334 A135335 A135336

nonn,tabf

N. J. A. Sloane, Dec 07 2007

Edited and extended by Emeric Deutsch, Dec 13 2007

approved