A209288 -id:A209288 - cp's OEIS frontend

A210472 Number A(n,k) of paths starting at {n}^k to a border position where one component equals 0 using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 33, 20, 1, 0, 1, 5, 196, 543, 70, 1, 0, 1, 6, 1305, 22096, 10497, 252, 1, 0, 1, 7, 9786, 1304045, 3323092, 220503, 924, 1, 0, 1, 8, 82201, 106478916, 1971644785, 574346824, 4870401, 3432, 1, 0

Offset: 0

Alois P. Heinz, Jan 22 2013

			A(0,3) = 1: [(0,0,0)].
A(1,1) = 1: [(1), (0)].
A(1,2) = 2: [(1,1), (0,1)], [(1,1), (1,0)].
A(1,3) = 3: [(1,1,1), (0,1,1)], [(1,1,1), (1,0,1)], [(1,1,1), (1,1,0)].
A(2,1) = 1: [(2), (1), (0)].
A(2,2) = 6: [(2,2), (1,2), (0,2)], [(2,2), (1,2), (1,1), (0,1)], [(2,2), (1,2), (1,1), (1,0)], [(2,2), (2,1), (1,1), (0,1)], [(2,2), (2,1), (1,1), (1,0)], [(2,2), (2,1), (2,0)].
Square array A(n,k) begins:
  0, 1,   1,      1,         1,             1, ...
  0, 1,   2,      3,         4,             5, ...
  0, 1,   6,     33,       196,          1305, ...
  0, 1,  20,    543,     22096,       1304045, ...
  0, 1,  70,  10497,   3323092,    1971644785, ...
  0, 1, 252, 220503, 574346824, 3617739047205, ...

Alois P. Heinz, Antidiagonals n = 0..24, flattened

Columns k=0-4 give: A000004, A000012, A000984, A209245, A209288.
Rows n=0-3 give: A057427, A001477, A093964, A210486.
Main diagonal gives A276490.
Cf. A089759 (unrestricted paths), A225094, A262809, A263159.

b:= proc() option remember; `if`(nargs=0, 0, `if`(args[1]=0, 1,
      add(b(sort(subsop(i=args[i]-1, [args]))[]), i=1..nargs)))
    end:
A:= (n, k)-> b(n$k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[] = 0; b[args__] := b[args] = If[First[{args}] == 0, 1, Sum[b @@ Sort[ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]]; a[n_, k_] := b @@ Array[n&, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

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}.

Original entry on oeis.org

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

cp's OEIS Frontend

A210472 Number A(n,k) of paths starting at {n}^k to a border position where one component equals 0 using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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