A286918 Total number of nodes summed over all lattice paths from (0,0) to (n,n) using steps {(k,0), (0,k) | 0

1, 3, 21, 159, 1257, 10046, 81811, 674184, 5605141, 46920874, 394949193, 3339464105, 28343082002, 241324470723, 2060357315568, 17632454524499, 151211115930880, 1299121121946684, 11179500360964780, 96344672766997340, 831385476662968094, 7182777882757416692

Offset: 0

Author

Alois P. Heinz, May 17 2017

Keywords

• nonn

Links

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

Crossrefs

Cf. A175891.

Programs

Maple

b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, [1$2], add((p-> p+[0, p[1]])(
         b(x-j, y)+b(x, y-j)), j=1..4)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..25);

Mathematica

b[x_, y_] := b[x, y] = If[y > x || y < 0, {0, 0}, If[x == 0, {1, 1}, Sum[Function[p, p + {0, p[[1]]}][b[x-j, y] + b[x, y-j]], {j, 1, 4}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / sqrt(n), 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 = 0.43633259077568249345422000202799136319250347607927734138960545201547... - Vaclav Kotesovec, May 30 2017