A108433 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k hills of the form ud (a hill is either a ud or a Udd starting at the x-axis).
1, 1, 1, 7, 2, 1, 47, 15, 3, 1, 361, 108, 24, 4, 1, 2977, 865, 184, 34, 5, 1, 25775, 7334, 1533, 276, 45, 6, 1, 231103, 64767, 13359, 2387, 385, 57, 7, 1, 2127409, 589368, 120376, 21368, 3450, 512, 70, 8, 1, 19990241, 5488033, 1112424, 196484, 31706
Offset: 0
Examples
Example T(2,1)=2 because we have udUdd and Uddud. Triangle begins: 1; 1,1; 7,2,1; 47,15,3,1; 361,108,24,4,1;
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
Programs
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Maple
A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=1/(1-z*A+z-t*z-z*A^2): Gserz:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 10 do P[n]:=sort(coeff(Gserz,z^n)) od: for n from 0 to 9 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form # second Maple program: b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0, `if`(x=0, 1, b(x-1, y-1, t)*`if`(t and y=1, z, 1)+ b(x-1, y+2, false)+b(x-2, y+1, is(y=0))))) end: T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(3*n, 0, false)): seq(T(n), n=0..10); # Alois P. Heinz, Oct 06 2015 -
Mathematica
b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0 || y > x, 0, If[x == 0, 1, b[x - 1, y - 1, t]*If[t && y == 1, z, 1] + b[x - 1, y + 2, False] + b[x - 2, y + 1, y == 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][b[3*n, 0, False]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 29 2016, after Alois P. Heinz *)
Formula
G.f.: 1/(1-tz+z-zA-zA^2), where A=1+zA^2+zA^3 or, equivalently, A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
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