I need to find the generating function of the following problem:
$d_n$ (for every natural number $n$) is the number of combinations to put coins into an automatic machine whereas the sum of the coins is $n$. There are coins of 1,10 and 25 cents and the amount of each coin is not limited.
While I can find the generating function when the order in which the coins are put into the machine doesn't matter, I don't have any idea in the case when the order do matter.
I know that it can be solved with exponential generating functions, but I wonder if there is any solution with "regular" generating functions.
If Im not wrong I think that the following
$$d_n=[x^n]\sum_{k=1}^\infty(x+x^{10}+x^{25})^k=[x^n]\frac{x+x^{10}+x^{25}}{1-(x+x^{10}+x^{25})}$$
count the ordered ways to put coins in the machine up to $n$.