A108440 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 having k u=(2,1) steps among the steps leading to the first d step.

1, 1, 1, 5, 4, 1, 33, 25, 7, 1, 249, 184, 54, 10, 1, 2033, 1481, 446, 92, 13, 1, 17485, 12620, 3863, 846, 139, 16, 1, 156033, 111889, 34637, 7881, 1411, 195, 19, 1, 1431281, 1021424, 318812, 74492, 14102, 2168, 260, 22, 1, 13412193, 9536113, 2995228

Offset: 0

Author

Emeric Deutsch, Jun 08 2005

Keywords

• nonn
• tabl

Examples

			T(2,1)=4 because we have udud, udUdd, uUddd and Uuddd.
Triangle begins:
.1;
.1,1;
.5,4,1;
.33,25,7,1;
.249,184,54,10,1;
		

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.

Crossrefs

Row sums yield A027307. Column 0 yields A034015.

Cf. A027307, A034015, A108441.

Programs

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-t*z*A-z*A^2): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 9 do P[n]:=coeff(Gser,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, false)+b(x-1, y+2, t)+
       b(x-2, y+1, t)*`if`(t, z, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(3*n, 0, true)):
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, False] + b[x - 1, y + 2, t] + b[x - 2, y + 1, t]*If[t, z, 1]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][ b[3*n, 0, True]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 29 2016, after Alois P. Heinz *)

Formula

G.f.: G(t, z)=1/(1-tzA-zA^2)-1, where A=1+zA^2+zA^3=(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).