O14-901
> restart:
> a:=arcsin:for i from 1 to 5 do a:=D(a) od;

                                       1
                        a := a -> ------------
                                            2
                                  sqrt(1 - a )


                                       a
                       a := a -> -------------
                                       2 (3/2)
                                 (1 - a )


                                2
                               a                1
              a := a -> 3 ------------- + -------------
                                2 (5/2)         2 (3/2)
                          (1 - a )        (1 - a )


                                3
                               a                  a
             a := a -> 15 ------------- + 9 -------------
                                2 (7/2)           2 (5/2)
                          (1 - a )          (1 - a )


                       4                  2
                      a                  a                  1
   a := a -> 105 ------------- + 90 ------------- + 9 -------------
                       2 (9/2)            2 (7/2)           2 (5/2)
                 (1 - a )           (1 - a )          (1 - a )


> # Conjecture : arcsin^(n)(a)=sum(b_(n,k) a^(n-2k-1)(1-a^2)^(-n+k+1/2),k=0..floor((n+1)/2)), le b_(n,k) étant des entiers positifs.
> c:= a^(q)*(1-a^2)^(-p-1/2);

                            q       2 (-p - 1/2)
                      c := a  (1 - a )

> d:=factor(diff(c,a));

          q                   (-p - 1/2)          2      2      2
         a  (-(a - 1) (a + 1))           (-q + q a  - 2 a  p - a )
    d := ---------------------------------------------------------
                             a (a - 1) (a + 1)

> # ce qui prouve l'hérédité
> 
>