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A260774
Certain directed lattice paths.
3
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 30 2015
EXTENSIONS
More terms from Lars Blomberg, Aug 01 2015
STATUS
approved