### Does there exist a generating function for the rational numbers?

Since the rationals are countable, you can list them in a sequence $(a_n)_{n\geq 0}$ such that each rational appears at least once in the sequence. Is there such a listing $(a_n)_{n \geq 0}$ for which $$\sum_{k = 0}^{\infty} a_kx^k =a_0 + a_1x + a_2x^2 + ...$$has a closed form?...Read more

### Generating Function, and Combinations

How many ways are there to obtain an even sum when 10 indistinguishable disce are rolled? hint: let $x_i$ be the number of dice showing the number $i$. OK, so the answer is $(\frac{1}{1-z})^3 [\binom{3}{1}(\frac{1}{1-z^2})(\frac{z}{1-z^2})^2+(\frac{1}{1-z^2})^3]$This is the same as saying $x_1+x_2+x_3+x_4+x_5+x_6=10$ with the condition that $x_1+2x_2+3x_3+4x_4+5x_5+6x_6$ is even. This is what the text says. This is what I understand so far$(\frac{1}{1-z})^3$ this is the $2x_2+4x_4+6x_6$ since these will always be even, but in reality they are ...Read more