A191314 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n and height k.

1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 7, 2, 1, 12, 6, 1, 1, 20, 12, 2, 1, 33, 27, 8, 1, 1, 54, 53, 16, 2, 1, 88, 108, 44, 10, 1, 1, 143, 208, 88, 20, 2, 1, 232, 405, 208, 65, 12, 1, 1, 376, 768, 415, 130, 24, 2, 1, 609, 1459, 908, 350, 90, 14, 1, 1, 986, 2734, 1804, 700, 180, 28, 2, 1, 1596, 5117, 3776, 1700, 544, 119, 16, 1

Offset: 0

Author

Emeric Deutsch, May 31 2011

Keywords

• nonn
• tabf

Comments

Row n has 1 + floor(n/2) entries.

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

T(n,1) = A000071(n+1) (Fibonacci numbers minus 1).

Sum_{k>=0} k * T(n,k) = A191315(n).

Extracting the even numbered rows, we obtain triangle A205946 with row sums A000984. The odd numbered rows yield triangle A205945 with row sums A001700. - Gary W. Adamson, Feb 01 2012

Examples

			T(5,2) = 2 because we have HUUDD and UUDDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
1;
1;
1,  1;
1,  2;
1,  4,  1;
1,  7,  2;
1, 12,  6, 1;
1, 20, 12, 2;
1, 33, 27, 8, 1;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A001405, A000071, A191315.

Cf. A205573, A205945, A001700, A205946, A000984.

Programs

Maple

F[0] := 1: F[1] := 1-z: for k from 2 to 12 do F[k] := sort(expand(F[k-1]-z^2*F[k-2])) end do: for k from 0 to 11 do h[k] := z^(2*k)/(F[k]*F[k+1]) end do: T := proc (n, k) options operator, arrow: coeff(series(h[k], z = 0, 20), z, n) end proc: for n from 0 to 16 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y) option remember; `if`(y>x or y<0, 0, `if`(x=0, 1,
      (p->add(coeff(p, z, i)*z^max(i, y), i=0..degree(p,z)))
      (b(x-1, y-1))+ b(x-1, y+1)+`if`(y=0, b(x-1, y), 0)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..14);  # Alois P. Heinz, Mar 12 2014

Mathematica

b[x_, y_] := b[x, y] = If[y>x || y<0, 0, If[x==0, 1, Function [{p}, Sum[ Coefficient[p, z, i]*z^Max[i, y], {i, 0, Exponent[p, z]}]][b[x-1, y-1]] + b[x-1, y+1] + If[y==0, b[x-1, y], 0]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

Formula

G.f.: The g.f. of column k is z^{2k}/(F[k]*F[k+1]), where F[k] are polynomials in z defined by F[0]=1, F[1]=1-z, F[k]=F[k-1]-z^2*F[k-2] for k>=2. The coefficients of these polynomials form the triangle A108299.

Rows may be obtained by taking finite differences of A205573 columns from the top -> down. - Gary W. Adamson, Feb 01 2012