cp's OEIS Frontend

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

%I A316673 #20 Nov 28 2024 15:11:34
%S A316673 1,13,818,64324,5592968,515092048,49239783968,4831678931008,
%T A316673 483371425775744,49083260519243008,5043379069021557248,
%U A316673 523221884090930480128,54715789513061864081408,5760456190025868833542144,609948004367577499751948288,64905519628343663567453569024
%N A316673 Number of paths from (0,0,0) to (n,n,n) that always move closer to (n,n,n).
%H A316673 Alois P. Heinz, <a href="/A316673/b316673.txt">Table of n, a(n) for n = 0..487</a>
%F A316673 Recurrence: see Maple program.
%F A316673 a(n) = A126086(n) * ceiling(2^(n-1)) = A126086(n) * A011782(n).
%F A316673 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
%F A316673 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
%p A316673 a:= proc(n) option remember; `if`(n<4, [1, 13, 818, 64324][n+1],
%p A316673      (2*(3*n-2)*(57*n^2-95*n+25)*a(n-1)-4*(9*n^3-30*n^2+29*n-6)*
%p A316673       a(n-2)+8*(3*n-1)*(n-2)^2*a(n-3))/(n^2*(3*n-4)))
%p A316673     end:
%p A316673 seq(a(n), n=0..20);
%t A316673 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))];
%t A316673 a /@ Range[0, 20] (* _Jean-François Alcover_, May 14 2020, after Maple *)
%Y A316673 Column k=3 of A316674.
%Y A316673 Cf. A052141, A126086.
%K A316673 nonn,walk
%O A316673 0,2
%A A316673 _Alois P. Heinz_, Jul 10 2018