A006347 a(n) = (n+1) a(n-1) + (-1)^n. (Formerly M3018)

0, 1, 3, 16, 95, 666, 5327, 47944, 479439, 5273830, 63285959, 822717468, 11518044551, 172770668266, 2764330692255, 46993621768336, 845885191830047, 16071818644770894, 321436372895417879, 6750163830803775460

Offset

1,3

Comments

a(n) is a function of the subfactorials .. a(n) = (n+1)!/2 - A000166(n+1) - Gary Detlefs, Apr 16 2010

a(n) can be seen indeed as a shifted-forward version of the subfactorials (or derangement numbers). - Olivier Gérard, Feb 23 2015

References

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

Links

Indranil Ghosh, Table of n, a(n) for n = 1..448

J. A. Sharp & N. J. A. Sloane, Correspondence, 1977

Formula

E.g.f.: x(1 - x/2 - exp(-x))/(1-x)^2.

a(n) = round((1/2 - exp(-1))*(n+1)!). - Benoit Cloitre, Sep 24 2006

a(n) = n(a(n-1) + a(n-2)), n > 2. - Gary Detlefs, Apr 10 2010

a(n) = 1/2*(n+1)! - floor(((n+1)!+1)/e). - Gary Detlefs, Apr 16 2010

Example

a(2) = (1/2)*6 - 2 = 1, a(3) = (1/2)*24 - 9 = 3, a(4) = (1/2)*120 - 44 = 16 ... - Gary Detlefs, Apr 16 2010

Maple

a:=n->-n!*sum((-1)^k/k!, k=3..n): seq(a(n), n=2..21); # Zerinvary Lajos, May 25 2007
seq(1/2*(n+1)! -floor(((n+1)!+1)/e), n=1..30); # Gary Detlefs, Apr 16 2010

Mathematica

RecurrenceTable[{a[1] == 0, a[n] == (n + 1) a[n - 1] + (-1)^n}, a, {n, 20}] (* Harvey P. Dale, Oct 19 2012 *)

Prog

(PARI) a(n)=if(n<2, 0, (n+1)*a(n-1)+(-1)^n)
(PARI) a(n)=round((1/2-exp(-1))*(n+1)!) \\ Benoit Cloitre, Sep 24 2006

Crossrefs

Cf. A000166.

Sequence in context: A137644 A114174 A181067 * A000270 A157051 A000271

Adjacent sequences: A006344 A006345 A006346 * A006348 A006349 A006350

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved