Question: stepwise solution of a summation

I've been trying to work out an algebraic expression for the following summation

S[n,t] = Sum(factorial(k)/factorial(k+t), k = 0 .. n-1), entered as S[n,t]:=sum(f,k=0..n-1); (with f:=k!/(k+t)!;)

I entered the above Maple expression into Maple (13) just so I could check out some small example summations I had. Maple's arithmetic, accuracy and speed is much superior to mine!

However, when I enterd the above expression, Maple gave me back a formula, the very one I had been trying to work out. (Nice bonus :) ). It is,

S[n,t] = -(n+t)*factorial(n)/((t-1)*factorial(n+t))+t/((t-1)*factorial(t))

All I need now is to find out how Maple arrived at that formula.

I know that Maple can do some stepwise solutions for calculus, linear algebra, vector calculus, but can it do me a stepwise solution of a summation?

I would dearly love to know how Maple arrived at that formula from the summation I gave it.

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