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A339390 Number of paths from (0,0,0) to (n,n,n) using steps (1,0,0), (0,1,0), (0,0,1), (1,1,1), and (2,2,2).

%I A339390 #38 May 30 2022 08:07:04
%S A339390 1,7,116,2397,54845,1329644,33464881,864627351,22776683200,
%T A339390 609024723535,16478750543705,450190397799036,12397538372467109,
%U A339390 343712858468053319,9584085091610235280,268571959802603851989,7558772037473679862681,213548821612723752662596
%N A339390 Number of paths from (0,0,0) to (n,n,n) using steps (1,0,0), (0,1,0), (0,0,1), (1,1,1), and (2,2,2).
%C A339390 The ratio of any two consecutive terms of this sequence a(n+1)/a(n) seems to grow asymptotically to ~30 as n increases (observation).
%H A339390 Alois P. Heinz, <a href="/A339390/b339390.txt">Table of n, a(n) for n = 0..679</a>
%F A339390 From _Alois P. Heinz_, Dec 05 2020: (Start)
%F A339390 a(n) = [(x*y*z)^n] 1/(1-x-y-z-x*y*z-(x*y*z)^2).
%F A339390 a(n) = ((3*n-7)*(3*n-2)*(30*n^2-50*n+13)*a(n-1) - (3*n-2)*(3*n-5)*a(n-2) - (45*n^4-300*n^3+677*n^2-560*n+108)*a(n-3) + (3*n-2)*(3*n-11)*a(n-4) + (3*n-1)*(9*n^3-75*n^2+197*n-154)*a(n-5) + (3*n-1)*(3*n-4)*(n-4)^2*a(n-6)) / ((3*n-4)*(3*n-7)*n^2) for n>=6. (End)
%p A339390 b:= proc(l) option remember; `if`(l[3]=0, 1,
%p A339390       add((f-> `if`(f[1]<0, 0, b(f)))(sort(l-h)), h=
%p A339390       [[1, 0$2], [0, 1, 0], [0$2, 1], [1$3], [2$3]]))
%p A339390     end:
%p A339390 a:= n-> b([n$3]):
%p A339390 seq(a(n), n=0..20);  # _Alois P. Heinz_, Dec 04 2020
%p A339390 # second Maple program:
%p A339390 a:= proc(n) local t; 1/(1-x-y-z-x*y*z-(x*y*z)^2);
%p A339390       for t in [x, y, z] do coeftayl(%, t=0, n) od
%p A339390     end:
%p A339390 seq(a(n), n=0..20);  # _Alois P. Heinz_, Dec 05 2020
%p A339390 # third Maple program:
%p A339390 a:= proc(n) option remember; `if`(n<6, [1, 7, 116, 2397, 54845,
%p A339390       1329644][n+1], ((3*n-7)*(3*n-2)*(30*n^2-50*n+13)*a(n-1) -(3*n-2)
%p A339390       *(3*n-5)*a(n-2) -(45*n^4-300*n^3+677*n^2-560*n+108)*a(n-3)
%p A339390       +(3*n-2)*(3*n-11)*a(n-4) +(3*n-1)*(9*n^3-75*n^2+197*n-154)*a(n-5)
%p A339390       +(3*n-1)*(3*n-4)*(n-4)^2*a(n-6)) / ((3*n-4)*(3*n-7)*n^2))
%p A339390     end:
%p A339390 seq(a(n), n=0..20);  # _Alois P. Heinz_, Dec 05 2020
%t A339390 b[l_] := b[l] = If[l[[3]] == 0, 1,
%t A339390      Sum[Function[f, If[f[[1]] < 0, 0, b[f]]][Sort[l-h]], {h,
%t A339390      {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 1, 1}, {2, 2, 2}}}]];
%t A339390 a[n_] := b[{n, n, n}];
%t A339390 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 30 2022, after _Alois P. Heinz_ *)
%Y A339390 Cf. A006480, A081798, A126086, A268550, A339565.
%K A339390 nonn
%O A339390 0,2
%A A339390 _William J. Wang_, Dec 02 2020