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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..679
Programs
-
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 *)
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)
Comments