A001120 a(0) = a(1) = 1; for n > 1, a(n) = n*a(n-1) + (-1)^n.

1, 1, 3, 8, 33, 164, 985, 6894, 55153, 496376, 4963761, 54601370, 655216441, 8517813732, 119249392249, 1788740883734, 28619854139745, 486537520375664, 8757675366761953, 166395831968477106, 3327916639369542121, 69886249426760384540, 1537497487388728459881

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 1..100 from T. D. Noe)
• J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Crossrefs

A000166 has a similar recurrence.

Column k=1 of A334715.

Cf. A000142.

Programs

Maple

a:= proc(n) option remember;
     `if`(n<2, 1, n*a(n-1)+(-1)^n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 09 2020

Mathematica

t = {1}; Do[AppendTo[t, n*t[[-1]] + (-1)^n], {n, 2, 20}]; t (* T. D. Noe, Jun 26 2012 *)
RecurrenceTable[{a[1]==1,a[n]==n a[n-1]+(-1)^n},a,{n,30}] (* Harvey P. Dale, Feb 19 2018 *)

PARI

a(n)=if(n<2,n>0,n*a(n-1)+(-1)^n)

PARI

a(n)=if(n<1,0,n!*polcoeff((1+exp(-x+x*O(x^n)))/(1-x),n))

Formula

Nearest integer to n!(1+1/e).

a(n) = A000166(n) + [n > 0] * n!. - Philippe Deléham, Sep 04 2005

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

E.g.f.: (exp(x)*x+1)*exp(-x)/(1-x). - Alois P. Heinz, May 08 2020

Extensions

a(0)=1 prepended by Alois P. Heinz, May 08 2020