A110043 -id:A110043 - cp's OEIS frontend

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

Alois P. Heinz, May 08 2020

			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, ...
  ...

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

Columns k=0-3 give: A000166, A001120, A110043, A110149.
Rows n=0-3 give: A000012, A001477, A005408, A016933.
Main diagonal gives A334716.
Cf. A000142.

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);

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 *)

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.

A110149 a(0) = 1, a(1) = 3; for n>1, a(n) = n*a(n-1) + (-1)^n.

Original entry on oeis.org

1, 3, 7, 20, 81, 404, 2425, 16974, 135793, 1222136, 12221361, 134434970, 1613219641, 20971855332, 293605974649, 4404089619734, 70465433915745, 1197912376567664, 21562422778217953, 409686032786141106, 8193720655722822121, 172068133770179264540

Offset: 0

Philippe Deléham, Sep 04 2005

A000166, A001120 and A110043 have a similar recurrence.

a(n) = (n-1)*(a(n-1)+a(n-2)), n>2. - Gary Detlefs, Apr 11 2010

Alois P. Heinz, Table of n, a(n) for n = 0..450

Column k=3 of A334715.

a:= proc(n) option remember;
      `if`(n<2, 2*n+1, n*a(n-1)+(-1)^n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 07 2020

RecurrenceTable[{a[1]==3,a[n]==n a[n-1]+(-1)^n},a,{n,20}]  (* Harvey P. Dale, Nov 21 2011 *)

a(n) = A110043(n) + n! = A001120(n) + 2*n! = A000166(n) + 3*n! for n>0.
a(n) = 3*n! + floor((n!+1)/e) for n>0. - Gary Detlefs, Apr 11 2010
E.g.f.: (3*exp(x)*x+1)*exp(-x)/(1-x). - Alois P. Heinz, May 07 2020

a(0)=1 prepended by Alois P. Heinz, May 07 2020

cp's OEIS Frontend

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.

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