OFFSET
0,2
COMMENTS
Level sums are defined as the sum of x(i,j,k) with i,j,k >= 0 and i+j+k = n. This gives 3*A164039(n-1) for n>0.
Slice x(1,j,k) with j,k >= 0 of the cube begins:
1, 1, 1, 1, 1, 1, 1, 1, ... A000012
1, 3, 5, 7, 9, 11, 13, 15, ... A005408
1, 5, 11, 19, 29, 41, 55, 71, ... A028387
1, 7, 19, 39, 69, 111, 167, 239, ... A108766(k+1)
1, 9, 29, 69, 139, 251, 419, 659, ...
1, 11, 41, 111, 251, 503, 923, 1583, ...
1, 13, 55, 167, 419, 923, 1847, 3431, ...
1, 15, 71, 239, 659, 1583, 3431, 6863, ...
The main diagonal of the slice is A134760.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..300
FORMULA
a(n) = x(n,n,n) with x(i,j,k) = 1 if 0 in {i,j,k} and x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) else.
a(n) ~ 3^(3*n+1/2) / (8*Pi*n). - Vaclav Kotesovec, Sep 07 2014
MAPLE
a:= proc(n) option remember; `if`(n<2, 2*n+1,
((888-3020*n+3668*n^2-1912*n^3+364*n^4) *a(n-1)
+3*(3*n-4)*(7*n-5)*(2*n-3)*(3*n-5) *a(n-2)) /
((2*n-1)*(7*n-12)*(n-1)^2))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Jan 17 2013
MATHEMATICA
b[] = 0; b[args__] := b[args] = If[{args}[[1]] == 0, 1, Sum[b @@ Sort[ ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]];
a[n_] := b @@ Table[n, 3];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 03 2018, from Alois P. Heinz's Maple code for A210472 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon Perry, Jan 13 2013
STATUS
approved