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

0,2

Links

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

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

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 *)

Crossrefs

Column k=3 of A316674.

Cf. A052141, A126086.

Sequence in context: A221934 A328033 A366559 * A319509 A189446 A296803

Adjacent sequences: A316670 A316671 A316672 * A316674 A316675 A316676

Keyword

nonn,walk

Author

Alois P. Heinz, Jul 10 2018

Status

approved