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).

1, 7, 116, 2397, 54845, 1329644, 33464881, 864627351, 22776683200, 609024723535, 16478750543705, 450190397799036, 12397538372467109, 343712858468053319, 9584085091610235280, 268571959802603851989, 7558772037473679862681, 213548821612723752662596

Offset

0,2

Comments

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..679

Formula

From Alois P. Heinz, Dec 05 2020: (Start)

a(n) = [(x*y*z)^n] 1/(1-x-y-z-x*y*z-(x*y*z)^2).

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)

Maple

b:= proc(l) option remember; `if`(l[3]=0, 1,
      add((f-> `if`(f[1]<0, 0, b(f)))(sort(l-h)), h=
      [[1, 0$2], [0, 1, 0], [0$2, 1], [1$3], [2$3]]))
    end:
a:= n-> b([n$3]):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 04 2020
# second Maple program:
a:= proc(n) local t; 1/(1-x-y-z-x*y*z-(x*y*z)^2);
      for t in [x, y, z] do coeftayl(%, t=0, n) od
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 05 2020
# third Maple program:
a:= proc(n) option remember; `if`(n<6, [1, 7, 116, 2397, 54845,
      1329644][n+1], ((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))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 05 2020

Mathematica

b[l_] := b[l] = If[l[[3]] == 0, 1,
     Sum[Function[f, If[f[[1]] < 0, 0, b[f]]][Sort[l-h]], {h,
     {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 1, 1}, {2, 2, 2}}}]];
a[n_] := b[{n, n, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)

Crossrefs

Cf. A006480, A081798, A126086, A268550, A339565.

Sequence in context: A328813 A178297 A285394 * A251585 A320083 A329543

Adjacent sequences: A339387 A339388 A339389 * A339391 A339392 A339393

Keyword

nonn

Author

William J. Wang, Dec 02 2020

Status

approved