OFFSET
0,2
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.
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
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
EXAMPLE
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;
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Mar 14 2005
STATUS
approved