A110043 a(0) = 1, a(1) = 2; for n>1, a(n) = n*a(n-1) + (-1)^n.

1, 2, 5, 14, 57, 284, 1705, 11934, 95473, 859256, 8592561, 94518170, 1134218041, 14744834532, 206427683449, 3096415251734, 49542644027745, 842224948471664, 15160049072489953, 288040932377309106, 5760818647546182121, 120977191598469824540, 2661498215166336139881

Offset: 0

Author

Philippe Deléham, Sep 04 2005

Keywords

• nonn
• easy

Comments

A000166 and A001120 have a similar recurrence.

Links

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

Crossrefs

Cf. A000142, A000166, A001120.

Column k=2 of A334715.

Programs

Maple

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

Mathematica

a[n_] := Subfactorial[n] + 2 Boole[n > 0] n!;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 18 2022 *)
nxt[{n_,a_}]:={n+1,a(n+1)+(-1)^(n+1)}; Join[{1},NestList[nxt,{1,2},30][[;;,2]]] (* Harvey P. Dale, Apr 14 2025 *)

Formula

a(n) = A001120(n) + n! = A000166(n) + 2*n! for n>0.

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

a(n) = 2*n! + floor((n!+1)/e) for n>0. - Gary Detlefs, Apr 11 2010

E.g.f.: (2*exp(x)*x+1)*exp(-x)/(1-x). - Alois P. Heinz, May 07 2020

Extensions

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