A339565 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (1,2), (2,1).

1, 3, 17, 101, 627, 3999, 25955, 170571, 1131433, 7559301, 50795985, 342935689, 2324278669, 15804931797, 107775401349, 736723618773, 5046774983235, 34636814325087, 238114193665451, 1639378334244867, 11301978856210543, 78010917772099207, 539055832175992119

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1179 (first 101 terms from Kent Mei)

Formula

a(n) = [(x*y)^n] 1/(1-x-y-x*y-x*y^2-x^2*y). - Alois P. Heinz, Dec 09 2020

Maple

a:= proc(n) local t; 1/(1-x-y-x*y-(x*y^2)-(x^2*y));
      for t in [x, y] do coeftayl(%, t=0, n) od
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020
# second Maple program:
b:= proc(l) option remember; `if`(l[2]=0, 1,
      add((f-> `if`(f[1]<0, 0, b(f)))(sort(l-h)), h=
      [[1, 0], [0, 1], [1$2], [1, 2], [2, 1]]))
    end:
a:= n-> b([n$2]):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020
# third Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 3, 17][n+1],
      ((6*n-3)*a(n-1)+(7*n-7)*a(n-2)+(4*n-6)*a(n-3))/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2020

Mathematica

b[l_] := b[l] = If[l[[2]] == 0, 1,
     Sum[Function[f, If[f[[1]] < 0, 0, b[f]]][Sort[l - h]], {h,
     {{1, 0}, {0, 1}, {1, 1}, {1, 2}, {2, 1}}}]];
a[n_] := b[{n, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)

Crossrefs

Cf. A000984, A001850, A137635, A339390.

Sequence in context: A330626 A161940 A074565 * A241768 A370286 A054365

Adjacent sequences: A339562 A339563 A339564 * A339566 A339567 A339568

Keyword

nonn

Author

Kent Mei, Dec 08 2020

Status

approved