Generating function for number of positives integers solution of equation a+2b+4c=n, n integer.

We assume that we have a country's currency that contains three coins worth 1, 3, and 4. How many ways can we get an amount of $n$ using these three pieces? In others words what is the number of partitions of $n$ into $1, 3$, and $4$?

What is the answer if $n=10$?

Is there a formula for the expression of $a_n$ in the general case of $n$? This is equivalent to find the number $a_n$ of positives integers $(a,b,c)$ solution of equation $a+3b+4c=n$, with $n$ positive integer. It well known that the number is the coefficient $a_n$ of $x^n$ in the generating function $\left(1+x+x^2+x^3+\ldots\right)\left(1+x^3+x^6+x^9+\ldots\right)\left(1+x^4+x^8+x^{12}+\ldots\right)\dots$

What are the calculation methods?