Does there exist an infinite series with sequence {$a_k$}$ \implies f(n,k)$ s.t for some positive integer $n$: $$\sum \limits_{k=0}^{\infty}f(n,k) = \frac{(n)^2(n+1)^2}{4}$$the right hand side being the sum of sequence of cubes for some positive integer $n$. For instance: $$\sum \limits_{k=0}^{\infty}f(2,k) = 9$$ $$\sum \limits_{k=0}^{\infty}f(3,k) = 36$$