  Many conforming finite elements on squares and cubes are elegantly classified
into families by the language of finite element exterior calculus and presented
in the Periodic Table of the Finite Elements. Use of these elements varies,
based principally on the ease or difficulty in finding a "computational basis"
of shape functions for element families. The tensor product family,
$Q^-_r\Lambda^k$, is most commonly used because computational basis functions
are easy to state and implement. The trimmed and non-trimmed serendipity
families, $S^-_r\Lambda^k$ and $S_r\Lambda^k$ respectively, are used less
frequently because they are newer to the community and, until now, lacked a
straightforward technique for computational basis construction. This represents
a missed opportunity for computational efficiency as the serendipity elements
in general have fewer degrees of freedom than elements of equivalent accuracy
from the tensor product family. Accordingly, in pursuit of easy adoption of the
serendipity families, we present complete lists of computational bases for both
serendipity families, for any order $r\geq 1$ and for any differential form
order $0\leq k\leq n$, for problems in dimension $n=2$ or $3$. The bases are
defined via shared subspace structures, allowing easy comparison of elements
across families. We use and include code in SageMath to find, list, and verify
these computational basis functions.
