O15-C924
> restart:
> f:=exp(-x^2/2);

                                          2
                          f := exp(- 1/2 x )

> for n from 1 to 10 do p[n]:=simplify((-1)^n*exp(x^2/2)*diff(f,x$n)) od;

                              p[1] := x


                                         2
                           p[2] := -1 + x


                                          2
                         p[3] := x (-3 + x )


                                       2    4
                        p[4] := 3 - 6 x  + x


                                         2    4
                     p[5] := x (15 - 10 x  + x )


                                     2       4    6
                   p[6] := -15 + 45 x  - 15 x  + x


                                       2       4    6
                p[7] := x (-105 + 105 x  - 21 x  + x )


                                 2        4       6    8
              p[8] := 105 - 420 x  + 210 x  - 28 x  + x


                                  2        4       6    8
           p[9] := x (945 - 1260 x  + 378 x  - 36 x  + x )


                             2         4        6       8    10
       p[10] := -945 + 4725 x  - 3150 x  + 630 x  - 45 x  + x

> for n from 1 to 9 do [simplify(p[n+1]-x*p[n]+diff(p[n],x)),simplify(diff(p[n+1],x)-(n+1)*p[n])] od;

                                [0, 0]


                                [0, 0]


                                [0, 0]


                                [0, 0]


                                [0, 0]


                                [0, 0]


                                [0, 0]


                                [0, 0]


                                [0, 0]

> for n from 1 to 10 do int((p[n])^2*exp(-x^2/2),x=-infinity..infinity) od;

                           sqrt(2) sqrt(Pi)


                          2 sqrt(2) sqrt(Pi)


                          6 sqrt(2) sqrt(Pi)


                         24 sqrt(2) sqrt(Pi)


                         120 sqrt(2) sqrt(Pi)


                         720 sqrt(2) sqrt(Pi)


                        5040 sqrt(2) sqrt(Pi)


                        40320 sqrt(2) sqrt(Pi)


                       362880 sqrt(2) sqrt(Pi)


                       3628800 sqrt(2) sqrt(Pi)