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

0,3

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).

Links

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

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).

Example

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;
  ...

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

Crossrefs

Cf. A001405, A191388, A191389.

Sequence in context: A191308 A191399 A191316 * A191793 A191791 A132597

Adjacent sequences: A191384 A191385 A191386 * A191388 A191389 A191390

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 02 2011

Status

approved