A175891 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k)|0

1, 1, 5, 29, 185, 1226, 8553, 61642, 455337, 3429002, 26229691, 203237747, 1591820564, 12582288455, 100241042348, 804090987555, 6488942266564, 52644171729304, 429123506792664, 3512829202462126, 28866426741057006, 238031465396515626, 1969001793889730276

Offset: 0

Author

Eric Werley, Dec 05 2010

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000103, A122951, A175883, A286918.

Programs

Maple

b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, add(b(x-j, y)+b(x, y-j), j=1..4)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, May 16 2017

Mathematica

b[x_, y_] := b[x, y] = If[y > x || y < 0, 0, If[x == 0, 1, Sum[b[x - j, y] + b[x, y - j], {j, 1, 4}]]];
a[n_] := b[n, n];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 8.84734830841870961487278801886633962039798... is the real root of the equation 4 + 4*d - 8*d^2 - 8*d^3 + d^4 = 0 and c = 0.31736815701423989167651891084531024477617724724822148387263881713... - Vaclav Kotesovec, May 30 2017