A316673 Number of paths from (0,0,0) to (n,n,n) that always move closer to (n,n,n).

1, 13, 818, 64324, 5592968, 515092048, 49239783968, 4831678931008, 483371425775744, 49083260519243008, 5043379069021557248, 523221884090930480128, 54715789513061864081408, 5760456190025868833542144, 609948004367577499751948288, 64905519628343663567453569024

Offset: 0

Author

Alois P. Heinz, Jul 10 2018

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..487

Crossrefs

Column k=3 of A316674.

Cf. A052141, A126086.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1, 13, 818, 64324][n+1],
     (2*(3*n-2)*(57*n^2-95*n+25)*a(n-1)-4*(9*n^3-30*n^2+29*n-6)*
      a(n-2)+8*(3*n-1)*(n-2)^2*a(n-3))/(n^2*(3*n-4)))
    end:
seq(a(n), n=0..20);

Mathematica

a[n_] := a[n] = If[n < 4, {1, 13, 818, 64324}[[n+1]], (2(3n-2)(57n^2- 95n+25) a[n-1] - 4(9n^3-30n^2+29n-6) a[n-2] + 8(3n-1)(n-2)^2 a[n-3]) / (n^2 (3n-4))];
a /@ Range[0, 20] (* Jean-François Alcover, May 14 2020, after Maple *)

Formula

Recurrence: see Maple program.

a(n) = A126086(n) * ceiling(2^(n-1)) = A126086(n) * A011782(n).

a(n) ~ sqrt((6 + 5*2^(1/3) + 4*2^(2/3))/6) * (24*2^(2/3) + 30*2^(1/3) + 38)^n / (4*Pi*n). - Vaclav Kotesovec, May 14 2020

G.f.: (1+hypergeom([1/3, 2/3],[1],108*x/(1-2*x)^3)/(1-2*x))/2. - Mark van Hoeij, Nov 28 2024