Numerical Determination of Generating Functions from Recursion Relations

Question

Are there computer packages which calculate coefficients of generating functions, such as

$$D_n(q)=\sum_m d_{m,n}q^m= \frac{1}{\prod_{i=1}^n (1-q^i)^2} \text{ or}$$

$$S_d(q)=\sum_m s_{m,d}q^m = \frac{q^d}{\prod_{i=1}^d (1-q^i)} \; ?$$

The effort, to calculate all coefficients e.g. $d_{k,l}$ for $k$ and $l$ smaller than $N$ should grow with $N^3$, as they satisfy recursion relations such as

$$d_{m,n} = \sum_{k=0}^{[m/n]}(k+1)d_{m-k\cdot n,n-1}$$

which allow to express each of the $N^2$ coefficients as a sum of at most $N$ terms ($d_{m,1}=m+1$, $d_{0,0}=1$, $d_{m,0}=0$ for $m>0$).

Before I revive my rusty computer knowledge I would like to know whether computer packages such as GAP tackle my problem to solve recursion relations with given initial conditions, save the solutions to files and allow to read them in again if one wants to tackle the next case $N+1$.

Answer 1

Check out Bruno Salvy (et al's) gfun package.

Answer 2

In Maple you can do

series(mul((1-q^i)^(-2),i=1..10),q=0,1000)

to get $D_{10}(q)$ as far as $q^{999}$; this took 0.9 seconds on my laptop. Alternatively, you can do

d := proc(m,n)
 option remember;

 if n = 0 then
  if m = 0 then
   return 1;
  else
   return 0;
  fi;
 else
  return add((k+1)*d(m-k*n,n-1),k=0..floor(m/n));
 fi;
end:

This calculates $d(1000,10)$ in about 0.9 seconds again. It remembers all instances of $d(m,n)$ that it has already computed.