How to solve this recurrence relation using generating functions: $a_n = 10 a_{n-1}-25 a_{n-2} + 5^n\binom{n+2}2$? How can we solve the following recurrence relation using GF?
$a_n = 10 a_{n-1}-25 a_{n-2} + 5^n {n+2 \choose 2}$ , for each $n>2, a_0 = 1, a_1 = 15$
I think that most of it is pretty straightforward. What really concerns me is this part
$5^n{n+2 \choose 2}$ 
Problem further analyzed
After trying to create generating functions in the equation we end up in this 
$\sum_{n=2}^\infty a_n x^n = 10 \sum_{n=2}^\infty {a_{n-1}} \sum_{n=2}^\infty a_{n-2} x^n + \sum_{n=2}^\infty {n+2 \choose 2}5^nx^n$
Lets take this part:
$ \sum_{n=2}^\infty {n+2 \choose 2}5^nx^n = \sum_{n=2}^\infty \frac{(n+1)(n+2)}{2} 5^nx^n $ 
then?

$\sum_{n=0}^\infty {n+2 \choose 2}5^nx^n = \frac12\sum_{n=0}^\infty(n^2+3n+2)(5x)^n=\frac12[\sum_{n=0}^\infty n^2y^n+3\sum_{n=0}^\infty ny^n+2\sum_{n=0}^\infty y^n]$
where $y=5x$. You will need the following series sums:
$(1)\sum_{n=0}^\infty n^2y^n=\dfrac{y(1+y)}{(1-y)^3}\\(2)\sum_{n=0}^\infty ny^n=\dfrac y{(y-1)^2}$
