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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)).
4
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
0,2
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
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
Cf. A006318 (row sums), A104547 (first column).
Sequence in context: A111797 A122104 A216121 * A121632 A186361 A197365
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Mar 14 2005
STATUS
approved