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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)).

%I A104546 #19 Jan 19 2023 03:31:26
%S A104546 1,2,5,1,16,6,60,29,1,245,138,11,1051,670,84,1,4660,3319,562,17,21174,
%T A104546 16691,3536,184,1,98072,84864,21510,1628,24,461330,435048,128134,
%U A104546 12860,345,1,2197997,2244532,752486,94534,3865,32,10585173,11639558,4373658,661498,37265,585,1
%N 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)).
%C A104546 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).
%C A104546 Row n contains 1 + floor(n/2) terms.
%H A104546 Alois P. Heinz, <a href="/A104546/b104546.txt">Rows n = 0..200, flattened</a>
%F A104546 T(n, 0) = A104547(n).
%F A104546 Sum_{k=0..floor(n/2)} T(n, k) = A006318(n) (row sums).
%F A104546 G.f.: G = G(t,z) satisfies G = 1 + z*G + z*G*(G + (t-1)*z/(1-z)).
%F A104546 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
%e A104546 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.
%e A104546 Triangle starts:
%e A104546       1;
%e A104546       2;
%e A104546       5,     1;
%e A104546      16,     6;
%e A104546      60,    29,     1;
%e A104546     245,   138,    11;
%e A104546    1051,   670,    84,    1;
%e A104546    4660,  3319,   562,   17;
%e A104546   21174, 16691,  3536,  184,  1;
%e A104546   98072, 84864, 21510, 1628, 24;
%t A104546 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 *)
%Y A104546 Cf. A006318 (row sums), A104547 (first column).
%K A104546 nonn,tabf
%O A104546 0,2
%A A104546 _Emeric Deutsch_, Mar 14 2005