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

0,8

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Wikipedia, Derangement

Wikipedia, Iverson bracket

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.

Example

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, ...
  ...

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 *)

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.

Sequence in context: A251718 A182457 A368605 * A026105 A303868 A060475

Adjacent sequences: A334712 A334713 A334714 * A334716 A334717 A334718

Keyword

nonn,tabl

Author

Alois P. Heinz, May 08 2020

Status

approved