Two different problems with similar solutions Problem 1 :
Calculate the sum
$$S(n):=\sum_{s=0}^{\infty}s^nx^s$$
The solution is S(n)=$\frac{P(n)}{(1-x)^{n+1}}$, where the polynomials
satisfy the reccurence
P(0)=1 , P(1)=x , P(n+1) = x(1-x)P(n)'+x(n+1)P(n)
Problem 2 :
Calculate the probabilities that the sum of n random variables X~U[0,1]
is in the range [0,1],[1,2],... (The random variables are independent)
The coefficients of the polynomials are
        1
      1   1
    1   4   1
  1   11  11  1
1  26   66  26  1

The probabilities of problem 2 are
$$ 1 $$
$$ \frac{1}{2}  \frac{1}{2}$$
$$ \frac{1}{6}  \frac{4}{6}  \frac{1}{6}$$
$$ \frac {1}{24}  \frac{11}{24}  \frac{11}{24}  \frac{1}{24}$$
The pattern is apparently similar to that of problem 1.
Can it be proven that the same numbers occur also for greater values n ?
I tried the falting-theorem for the sum of random variables, but I could not
 prove completely, that the same values as in problem 1 occur.

This continues to be true.  The coefficients of your polynomials are the Eulerian Numbers (and the polynomials the Eulerian polynomials).
Compare the formula in terms of binomial coefficients for the Eulerian numbers given in the above link with the integral of the pdf of the Irwin-Hall Distribution between consecutive integers (you'll need to simplify a bit to get the identity).
