Example 5.3 Find $P_{4}(20)$.

 untranslated text remains the same as it is a mathematical expression.
Solution: $\begin{aligned} P_{4}(20)= & \sum_{k=1}^{\left[\frac{20}{4}\right]} P_{3}(20-4 k+4-1) \\ = & \sum_{k=1}^{5} P_{3}(23-4 k) \\ = & P_{3}(19)+P_{3}(15)+P_{3}(11)+P_{3}(7)+P_{3}(3) \\ = & {\left[\frac{19^{2}+3}{12}\right]+\left[\frac{15^{2}+3}{12}\right]+\left[\frac{11^{2}+3}{12}\right]+} \\ & {\left[\frac{7^{2}+3}{12}\right]+\left[\frac{3^{2}+3}{12}\right] } \\ = & 30+19+10+4+1=64 .\end{aligned}$