A104546 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k platforms (i.e., UHD, UHHD, UHHHD, ..., where U=(1,1), D=(1,-1), H=(2,0)).

1, 2, 5, 1, 16, 6, 60, 29, 1, 245, 138, 11, 1051, 670, 84, 1, 4660, 3319, 562, 17, 21174, 16691, 3536, 184, 1, 98072, 84864, 21510, 1628, 24, 461330, 435048, 128134, 12860, 345, 1, 2197997, 2244532, 752486, 94534, 3865, 32, 10585173, 11639558, 4373658, 661498, 37265, 585, 1

Offset: 0

Author

Emeric Deutsch, Mar 14 2005

Keywords

• nonn
• tabf

Comments

A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318).

Row n contains 1 + floor(n/2) terms.

Examples

			T(3,1) = 6 because we have H(UHD), UD(UHD), (UHD)H, (UHD)UD, (UHHD), U(UHD)D; the platforms are shown between parentheses.
Triangle starts:
      1;
      2;
      5,     1;
     16,     6;
     60,    29,     1;
    245,   138,    11;
   1051,   670,    84,    1;
   4660,  3319,   562,   17;
  21174, 16691,  3536,  184,  1;
  98072, 84864, 21510, 1628, 24;
		

Links

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

Crossrefs

Cf. A006318 (row sums), A104547 (first column).

Programs

Mathematica

With[{m=20}, CoefficientList[CoefficientList[Series[(1 -2*y +(2-x)*y^2 - Sqrt[1 -8*y +2*(8-x)*y^2 +4*(x-3)*y^3 +(x-2)^2*y^4])/(2*y*(1-y)), {y,0,m}, {x,0,Floor[m/2]}], y], x]]//Flatten (* G. C. Greubel, Jan 19 2023 *)

Formula

T(n, 0) = A104547(n).

Sum_{k=0..floor(n/2)} T(n, k) = A006318(n) (row sums).

G.f.: G = G(t,z) satisfies G = 1 + z*G + z*G*(G + (t-1)*z/(1-z)).

G.f.: (1 - 2*y + (2-x)*y^2 - sqrt(1 - 8*y + 2*(8-*x)*y^2 + 4*(x-3)*y^3 + (x-2)^2*y^4))/(2*y*(1-y)). - G. C. Greubel, Jan 19 2023