A191387 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k valleys at level 0.

1, 1, 2, 3, 5, 1, 8, 2, 14, 5, 1, 23, 10, 2, 41, 22, 6, 1, 69, 42, 13, 2, 125, 87, 32, 7, 1, 214, 164, 66, 16, 2, 393, 330, 149, 43, 8, 1, 682, 618, 301, 94, 19, 2, 1267, 1225, 648, 227, 55, 9, 1, 2223, 2288, 1290, 484, 126, 22, 2, 4171, 4498, 2700, 1100, 322, 68, 10, 1, 7385, 8396, 5322, 2300, 718, 162, 25, 2

Offset: 0

Author

Emeric Deutsch, Jun 02 2011

Keywords

• nonn
• tabf

Comments

A dispersed Dyck paths of length n is a Motzkin path of length n with no (1,0) steps at positive heights.

Row n >=2 has floor(n/2) entries.

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

T(n,0) = A191388(n).

Sum_{k>=0} k*T(n,k) = A191389(n).

Examples

			T(5,1)=2 because we have HUDUD and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
   1;
   1;
   2;
   3;
   5,  1;
   8,  2;
  14,  5,  1;
  23, 10,  2;
  41, 22,  6,  1;
  ...
		

Links

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

Crossrefs

Cf. A001405, A191388, A191389.

Programs

Maple

G := (1+z^2*c-t*z^2*c)/(1-z-z^3*c-t*z^2*c*(1-z)): c := ((1-sqrt(1-4*z^2))*1/2)/z^2: Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)-1) end do; # yields sequence in triangular form

Formula

G.f.: G=G(t,z) is given by G = 1 + z*G + z^2*c*(t*(G-1-z*G) + 1 + z*G), where c = (1-sqrt(1-4*z^2))/(2*z^2).