O17-960
> restart:
> alfa:=n->n!*sum((-1)^k/(k!),k=0..n);

                                    /  n        \
                                    |-----     k|
                                    | \    (-1) |
                    alfa := n -> n! |  )   -----|
                                    | /     k!  |
                                    |-----      |
                                    \k = 0      /

> seq(alfa(n),n=1..10);

          0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961

> seq(alfa(n+1)-(n+1)*alfa(n),n=1..10);

                  1, -1, 1, -1, 1, -1, 1, -1, 1, -1

Conjecture : alfa(n+1)-(n+1)*alfa(n)=(-1)^(n-1)
alfa(n) est le nombre entier le plus proche de n!/e
> ediff:=(1-x)*diff(y(x),x)-x*y(x);

                                  /d      \
                 ediff := (1 - x) |-- y(x)| - x y(x)
                                  \dx     /

> resol:=dsolve({ediff,y(0)=1},y(x));

                                        exp(-x)
                      resol := y(x) = - -------
                                        -1 + x

> f:=subs(resol,y(x));g:=unapply(f,x);

                                   exp(-x)
                            f := - -------
                                   -1 + x


                                     exp(-x)
                         g := x -> - -------
                                     -1 + x

> seq(((D@@n)(g))(0),n=1..10);

          0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961

Conjecture : beta(n)= alfa(n)
Preuve : beta(n+1)=nbeta(n)+nbeta(n-1) donne bien beta(n+1)-(n+1)beta(n)=(-1)^n
sum(k parmi n*alfa(k))=D^n(f*exp)(0) (Leibnitz)=n!
>