A334715 A(n,k) = !n + [n > 0] * (k * n!), where !n = A000166(n) is subfactorial of n and [] is an Iverson bracket; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 3, 5, 8, 9, 1, 4, 7, 14, 33, 44, 1, 5, 9, 20, 57, 164, 265, 1, 6, 11, 26, 81, 284, 985, 1854, 1, 7, 13, 32, 105, 404, 1705, 6894, 14833, 1, 8, 15, 38, 129, 524, 2425, 11934, 55153, 133496, 1, 9, 17, 44, 153, 644, 3145, 16974, 95473, 496376, 1334961

Offset: 0

Author

Alois P. Heinz, May 08 2020

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
     1,    1,     1,     1,     1,     1,     1,     1, ...
     0,    1,     2,     3,     4,     5,     6,     7, ...
     1,    3,     5,     7,     9,    11,    13,    15, ...
     2,    8,    14,    20,    26,    32,    38,    44, ...
     9,   33,    57,    81,   105,   129,   153,   177, ...
    44,  164,   284,   404,   524,   644,   764,   884, ...
   265,  985,  1705,  2425,  3145,  3865,  4585,  5305, ...
  1854, 6894, 11934, 16974, 22014, 27054, 32094, 37134, ...
  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Wikipedia, Derangement
• Wikipedia, Iverson bracket

Crossrefs

Columns k=0-3 give: A000166, A001120, A110043, A110149.

Rows n=0-3 give: A000012, A001477, A005408, A016933.

Main diagonal gives A334716.

Cf. A000142.

Programs

Maple

A:= proc(n, k) option remember; `if`(n<2,
      (k-1)*n+1, n*A(n-1, k)+(-1)^n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

A[n_, k_] := Subfactorial[n] + Boole[n>0] k n!;
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

Formula

E.g.f. of column k: (k*exp(x)*x+1)*exp(-x)/(1-x).

A(n,k) = A000166(n) + [n > 0] * (k * n!).

A(n,k) = (k-1)*n + 1 if n<2, A(n,k) = n*A(n-1, k) + (-1)^n if n>=2.