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
KEYWORD
nonn
AUTHOR
William J. Wang, Dec 02 2020
STATUS
approved