A108441 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=(1,2) steps among the steps leading to the first d step.
1, 1, 1, 3, 6, 1, 15, 39, 11, 1, 97, 284, 100, 16, 1, 721, 2249, 888, 186, 21, 1, 5827, 18890, 7977, 1952, 297, 26, 1, 49759, 165519, 72991, 19731, 3601, 433, 31, 1, 441729, 1496696, 680096, 196864, 40586, 5960, 594, 36, 1, 4035937, 13865297, 6439656
Offset: 0
Examples
T(2,1)=6 because we have uUddd, Uddud, UddUdd, Ududd, UdUddd and Uuddd. Triangle begins: 1; 1,1; 3,6,1; 15,39,11,1; 97,284,100,16,1;
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-t*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)* `if`(t, z, 1)+b(x-2, y+1, t)))) 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]*If[t, z, 1] + b[x-2, y+1, t]]]]; 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-zA-tzA^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).
Comments