Macsyma is a symbolic manipulation system with origins in the 1960s.
Download its current freeware implementation from
http://maxima.sourceforge.net/download.html
When you have installed it on your computer, you can do things like
obtain the formulas for recursively-defined function. For example,
the Chebyshev polynomials may be obtained by the commands
t[0,x]:1;
t[1,x]:x;
t[n,x]:= 2*x*t[n-1,x]-t[n-2,x];
The ":" assignment command evaluates the expression on the right
first, while the ":=" assignment command waits so that recursive
expressions may be expanded.
To print the first 21 Chebyshev polynomials, use
for n:0 thru 20 do print(expand(t[n,x]));