A260774 Certain directed lattice paths.

1, 6, 33, 189, 1107, 6588, 39663, 240894, 1473147, 9058554, 55954395, 346934745, 2157989445, 13459891500, 84152389833, 527224251861, 3309194474451, 20804569738218, 130987600581699, 825796890644895, 5212349717906889, 32935490120006604, 208316726580941037

Offset

0,2

Comments

See Dziemianczuk (2014) for precise definition.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1235 (first 101 terms from Lars Blomberg)

M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

Formula

See Dziemianczuk (2014) Equation (33a) with m=1.

From Vaclav Kotesovec, Jul 15 2022: (Start)

Recurrence: (n+1)*(4*n - 3)*a(n) = 6*(4*n^2 - n - 1)*a(n-1) + 3*(n-1)*(4*n + 1)*a(n-2).

a(n) ~ (3 + 2*sqrt(3))^(n+1) / sqrt(6*Pi*n). (End)

Maple

b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
      `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
      `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 21 2021

Mathematica

b[x_, y_] := b[x, y] = If[{x, y} == {0, 0}, 1,
     If[x > 0, Sum[b[x - 1, y + j], {j, -1, 1}], 0] +
     If[y > 0, b[x, y - 1], 0] + If[y < 0, b[x, y + 1], 0]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

Crossrefs

Cf. A122868, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

Sequence in context: A180035 A360717 A286187 * A218182 A093964 A361776

Adjacent sequences: A260771 A260772 A260773 * A260775 A260776 A260777

Keyword

nonn

Author

N. J. A. Sloane, Jul 30 2015

Extensions

More terms from Lars Blomberg, Aug 01 2015

Status

approved