I am asked to find the number of ways to give change of 20.19 dollars in pennies,nickels, dimes, quarters, half-dollars, and dollars... This is a classical problem and I can generate a function $ G(x) = \frac{1}{(1-x)(1-x^5)(1-x^{10})(1-x^{25})(1-x^{50})(1-x^{100})}$ and find the coefficient of $x^{2019}$ using a computer. But we are asked to do it by hand using functions and the formula $B(x) = C(x) \sum_k \left(\begin{array}{c}k+5\\ 5\end{array}\right)$ ( the proof is long and it can be found here on page 174: https://doc.lagout.org/science/0_Computer%20Science/3_Theory/Graph%20Theory/Combinatorics%20and%20Graph%20Theory.pdf ). This method is confusing me a lot. I tried to mimick the method without deep understanding and I arrived at $a_{2019} = b_{403} = c_0 \left (\begin{array}{c}25\\ 5\end{array}\right ) + c_ {23} \left (\begin{array}{c}24\\ 5\end{array}\right ) + c_{43} \left (\begin{array}{c}23\\ 5\end{array}\right ) + c_{63} \left (\begin{array}{c}22\\ 5\end{array}\right )$ but I cannot proceed because there's no $c_{83}$ since the highest degree is 81. Can someone please explain to me how this works?
There's a similar answer here at 2.6.3#1 https://www.math.ksu.edu/~gerald/math510/hw12.pdf but it is still not clear to me
Thank you very much!