A191320 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 HUs, where U=(1,1) and H=(1,0).

1, 1, 2, 2, 1, 4, 2, 4, 6, 9, 10, 1, 9, 23, 3, 23, 36, 11, 23, 77, 25, 1, 65, 118, 65, 4, 65, 249, 131, 17, 197, 380, 298, 48, 1, 197, 808, 566, 140, 5, 626, 1236, 1210, 336, 24, 626, 2665, 2230, 833, 80, 1, 2056, 4094, 4627, 1828, 259, 6, 2056, 8955, 8401, 4155, 711, 32, 6918, 13816, 17192, 8648, 1923, 122, 1

Offset: 0

Author

Emeric Deutsch, Jun 01 2011

Keywords

• nonn
• tabf

Comments

Row n has 1 + floor(n/3) entries.

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

T(2*n,0) = T(2*n+1,0) = A014137(n) (partial sums of the Catalan numbers).

Sum_{k>=0}k*T(n,k) = A093387(n).

Examples

			T(7,2)=3 because we have HHUDHUD, HUDHHUD, and HUDHUDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
   1;
   1;
   2;
   2,  1;
   4,  2;
   4,  6;
   9, 10,  1;
   9, 23,  3;
  23, 36, 11;
		

Crossrefs

Cf. A001405, A014137, A093387.

Programs

Maple

G := 2/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

Formula

G.f.: G(t,z) = 2/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)).