I just want to consider the simplest case:

Let S=[0,0,0,0,1], how to derive the general formula for $Sym^k$ S?

My conjectured formula based on the results of LiE program for finite k values is:

$Sym^k(S)=\overset{\left[k/2\right]}{\underset{i=0}{\oplus}}[i,0,0,0,k-2i]$

But I have no clue how to prove it or even a derivation for the simplest case when k=2.

Any reference or tips would be greatly appreciated.