A191308 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 having k peaks at height >=2.

1, 1, 2, 3, 5, 1, 8, 2, 13, 6, 1, 21, 12, 2, 34, 26, 9, 1, 55, 50, 19, 2, 89, 97, 52, 13, 1, 144, 180, 108, 28, 2, 233, 332, 243, 97, 18, 1, 377, 600, 488, 210, 39, 2, 610, 1076, 999, 552, 170, 24, 1, 987, 1908, 1942, 1168, 376, 52, 2, 1597, 3361, 3773, 2663, 1162, 282, 31, 1, 2584, 5878, 7140, 5466, 2543, 630, 67, 2

Offset: 0

Author

Emeric Deutsch, May 30 2011

Keywords

• nonn
• tabf

Comments

Also number of dispersed Dyck paths of length n having k valleys (i.e., DU's). Example: T(7,2)=2 because we have HUDUDUD and UDUDUDH, where U=(1,1), D=(1,-1), H=(1,0).

Also number of dispersed Dyck paths of length n having k doublerises (i.e., UU's). Example: T(7,2)=2 because we have HUUUDDD and UUUDDDH, where U=(1,1), D=(1,-1), H=(1,0).

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) = A000045(n+1) (the Fibonacci numbers).

T(n,1) = A054454(n-4).

Sum_{k>=0} k*T(n,k) = A191309(n).

Examples

			T(7,2)=2 because we have HUUDUDD and UUDUDDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
   1;
   1;
   2;
   3;
   5,  1;
   8,  2;
  13,  6,  1;
  21, 12,  2;
		

Crossrefs

Cf. A000045, A001405, A054454, A191309.

Programs

Maple

r := ((1-z^2-t*z^2-sqrt(((1-z)^2-t*z^2)*((1+z)^2-t*z^2)))*1/2)/(t*z^2): G := 1/(1-z-z^2*(1+t*r)): 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) satisfies G = 1+z*G + z^2*G(1+t*r), where r=r(t,z) is given by r=z^2*(1+r)*(1+t*r) (the Narayana function).