To answer your question about the generating function...
Counting the number of ways to make change for $\$n$ is equivalent to counting the number of integer solutions to $$ k_1+2k_2+5k_5+10k_{10}=n,\\\\\tag{*} k_1,k_2,k_5,k_{10}\ge 0 $$ Here, $k_i$ represents the number of $\$i$ bills. I claim that this is the coefficient of $x^n$ in $$ (1+x+x^2+\dots)(1+x^2+x^4+x^6+\dots)(1+x^5+x^{10}+\dots)(1+x^{10}+x^{20}+\dots)\tag {**} $$ This is because when expanding out the product of infinite sums in $(**)$, each term is a product of terms that look like $x^{k_1}(x^2)^{k_2}(x^5)^{k_5}(x^{10})^{k_{10}}$, so the number of summands equal to $x^n$ is the number of solutions to $(*)$.
Applying the geometric series formula to each factor in $(**)$, we get the generating function is $$ \boxed{\frac1{1-x}\cdot \frac1{1-x^2}\cdot \frac1{1-x^5}\cdot \frac1{1-x^{10}}} $$