Can I have formula for all natural numbers, if I know it separately for even and for odd? Solving one problem I find two formulas, first for odd:
$$f(m,n)=\frac{n^m-n}{m}+n$$
and second for even numbers:
$$f(m,n)=\frac{n^m-n^2}{m}+\frac{n(n+1)}{2}$$
Then for static $m$ and dynamic $n$ we have:
$$f(2,n)=1,3,6,10,15,21,28,36,44\cdots$$
$$f(3,n)=1,4,11,24,45,76,119,176\cdots$$
$$f(4,n)=1,6,24,70,165,336,616,1044\cdots$$
$$f(5,n)=1,8,51,208,629,1560,3367,6560\cdots$$
$$f(6,n)=1,14,130,700,2635,7827,19684\cdots$$
Can I have formula for all natural numbers, if I know it separately for even and for odd?
If I made some mistakes, sorry for my English.

$$f(m,n)=\frac{(-1)^{m+1}+1}{2}\left(\frac{n^m-n}{m}+n \right)+ 
\frac{(-1)^{m}+1}{2}\left(\frac{n^m-n^2}{m}+\frac{n(n+1)}{2} \right)$$
