A209245 Main diagonal of the triple recurrence x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) with x(i,j,k) = 1 if 0 in {i,j,k}.

1, 3, 33, 543, 10497, 220503, 4870401, 111243135, 2602452993, 61985744967, 1497148260033, 36566829737727, 901314269530113, 22385640256615743, 559574590912019457, 14065064484334380543, 355222860485671141377, 9008982166319523972903, 229325469394627488082497

Offset: 0

Author

Jon Perry, Jan 13 2013

Keywords

• nonn

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

Crossrefs

Cf. A164039, A134760, A209288.

Column k=3 of A210472. - Alois P. Heinz, Jan 23 2013

Programs

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 *)

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