A191384 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) with k ascents of length 1. An ascent is a maximal sequence of consecutive (1,1)-steps.

1, 1, 1, 1, 1, 2, 2, 3, 1, 3, 4, 3, 5, 8, 6, 1, 7, 14, 10, 4, 12, 26, 21, 10, 1, 18, 42, 41, 20, 5, 31, 77, 83, 45, 15, 1, 47, 128, 150, 96, 35, 6, 81, 234, 293, 209, 85, 21, 1, 125, 388, 530, 414, 196, 56, 7, 216, 704, 1023, 858, 455, 147, 28, 1, 337, 1172, 1828, 1668, 974, 364, 84, 8, 583, 2119, 3479, 3385, 2133, 896, 238, 36, 1

Offset: 0

Author

Emeric Deutsch, Jun 01 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,0) = A191385(n).

Sum_{k>=0} k*T(n,k) = A191386(n).

Examples

			T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
  1;
  1;
  1,  1;
  1,  2;
  2,  3,  1;
  3,  4,  3;
  5,  8,  6,  1;
  7, 14, 10,  4;
		

Crossrefs

Cf. A001405, A191385, A191386.

Programs

Maple

G := ((t*z^2-(1-z)^2+sqrt((1+z^2-t*z^2)*(1-3*z^2-t*z^2)))*1/2)/(z*(1-2*z+z^2-z^3-t*z^2+t*z^3)): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Formula

G.f.: G(t,z) = (t*z^2 - (1-z)^2 + sqrt((1+z^2-t*z^2)*(1-3*z^2-t*z^2)))/(2*z*(1-2*z+z^2-z^3-t*z^2+t*z^3)).