The Frobenius equation is the Diophantine equation $$ a_1 x_1+\dots+a_n x_n=b,$$ where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-negative integers, i.e. $$ x_j \in \mathbb{N} $$ as Natural numbers. For negative $b$, there are no solutions.

• My question: Is there any known formula that counts the number of solutions, by giving $a_1, \dots, a_n$, $b$, and $n$? Let us call this function as $F (a_1, \dots, a_n; b, n)$, what is known for this:

$$F (a_1, \dots, a_n; b, n)=?$$

For all the $a_j=1$, we can simplify the above Frobenius equation to: $$ x_1+\dots+x_n=b, \tag{1}$$ where $b \in \mathbb{Z}^+$ is a positive integer.

• Here is another simpler question: Is there a general formula for Eq.(1) counting all the possible solutions $$(x_1, \dots, x_n)$$
  for given the positive integer $n \in \mathbb{Z}^+$ and $b \in \mathbb{Z}^+$? This should be related to the Partition, but I am not sure the exact forms are known? Say, can we find the total number of soultions as a function $f(n,b)$, and what is $$ f(n,b)=? $$

It seems the answer is known:

$$ f(n,b)= \binom{b+n-1}{n-1}? $$

p.s. Sorry if this question is too simple for number theorists. But please provide me answer and Refs if you already know the answer. Many thanks!