Subset zeta transform ($F(S)={\sum }_{T\subseteq S}f(T)$)

The subset zeta transform sends a function $f$ on subsets to $F(S)={\sum }_{T\subseteq S}f(T)$. For subsets of an $n$-element set, all values can be computed in $O(n{2}^n)$ additions by processing one element at a time. Reversing containment gives the superset transform.