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

0,6

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

Links

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

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

Example

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;

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 *)

Crossrefs

Cf. A001405, A000071, A191315.

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

Sequence in context: A211232 A137633 A168533 * A191306 A191525 A066633

Adjacent sequences: A191311 A191312 A191313 * A191315 A191316 A191317

Keyword

nonn,tabf

Author

Emeric Deutsch, May 31 2011

Status

approved