> restart:
> f:=GAMMA(n+d+1)/GAMMA(n+1)/GAMMA(d+1):
> p:=convert(series(f,n,2),polynom):
> sol:=solve(p=k,n);
> with(plots):
> plot3d(f-p,n=0..5,d=0..5,axes=boxed);

                                   -1 + k
                      sol := ------------------
                             Psi(d + 1) + gamma


> assume(n,integer,d,integer);convert(f,factorial);

                              (n~ + d~)!
                              ----------
                               n~! d~!

> product((n+k)/k,k=1..d);

                         GAMMA(n~ + d~ + 1)
                     ---------------------------
                     GAMMA(n~ + 1) GAMMA(1 + d~)

> for i from 1 to 5 do
>   expand(subs(n=i,f));
> od;

                                1 + d~


                                            2
                         1 + 3/2 d~ + 1/2 d~


                                     2         3
                     1 + 11/6 d~ + d~  + 1/6 d~


                   25      35   2          3          4
               1 + -- d~ + -- d~  + 5/12 d~  + 1/24 d~
                   12      24


             137             2   17   3         4           5
         1 + --- d~ + 15/8 d~  + -- d~  + 1/8 d~  + 1/120 d~
             60                  24

> 
