A108766 -id:A108766 - cp's OEIS frontend

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

Jon Perry, Jan 13 2013

nonn

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.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A164039, A134760, A209288.

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

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

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 *)

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

A127737 A002260 * A127701.

Original entry on oeis.org

1, 3, 4, 3, 7, 9, 3, 7, 13, 16, 3, 7, 13, 21, 25, 3, 7, 13, 21, 31, 36, 3, 7, 13, 21, 31, 43, 49, 3, 7, 13, 21, 31, 43, 57, 64, 3, 7, 13, 21, 31, 43, 57, 73, 81, 3, 7, 13, 21, 31, 43, 57, 73, 91, 100

Offset: 1

Gary W. Adamson, Jan 27 2007

Deleting the right border (1, 4, 9, 16, ...), rows tend to A002061 starting (3, 7, 13, 21, 31, ...). Row sums = A108766: (1, 7, 19, 39, 69, 111, ...).

			First few rows of the triangle:
  1;
  3, 4;
  3, 7,  9;
  3, 7, 13, 16;
  3, 7, 13, 21, 25;
  ...

Cf. A002260, A127701, A108766, A002061.

A002260 * A127701 as infinite lower triangular matrices.

cp's OEIS Frontend

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}.

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