A105926 First differences of A000166.

-1, 1, 1, 7, 35, 221, 1589, 12979, 118663, 1201465, 13349609, 161530271, 2114578091, 29780308117, 448995414685, 7215997736011, 123153028027919, 2224451568754289, 42395429898611153, 850263899633257015, 17900292623858042419, 394701452356069835341

Offset: 0

Author

N. J. A. Sloane, Apr 27 2005

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000166.

Programs

Maple

a:=n->sum((-1)^k * (n-k-1) * n!/k!, k=0..n): seq(a(n), n=0..20); # Zerinvary Lajos, Jun 27 2007
A000166:= gfun:-rectoproc({a(0)=1,a(1)=0,a(n) = (n-1)*(a(n-1)+a(n-2))},a(n),remember):
seq(A000166(n+1)-A000166(n),n=0..100); # Robert Israel, Nov 03 2015

Mathematica

Table[Subfactorial[n] - Subfactorial[n - 1], {n, 1, 22}] (* Zerinvary Lajos, Jul 09 2009 *)
Table[n Subfactorial[n] - (-1)^n, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
Differences[Table[(-1)^n HypergeometricPFQ[{-n,1},{},1], {n,0,20}]] (* Peter Luschny, Nov 03 2015 *)

Formula

a(n) = n*!n - (-1)^n, where !n = A000166(n) is subfactorial. - Vladimir Reshetnikov, Nov 03 2015

(2n + 1) a(n+2) = (2n^2 + 5n + 4) a(n+1) + (2n^2 + 5n + 3) a(n). E.g.f.: exp(-x)*(2*x-1)/(x-1)^2. - Robert Israel, Nov 03 2015