Sum of $k$th power of first $n$ natural number (power sum)

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I was working on a problem which involves computation power sum (summation of $k^{th}$ power of first natural number), can someone help me how to simplify the below equation.

I can compute power sum $n^k$ efficiently.

Below is the given equation

$$f(0,n,k)=n^k$$ $$f(m,n,k)=\sum_{i=0}^n f(m-1,i,k)$$

Problem link

16686. Some Sums