  Sporadic Ap\'ery-like sequences were discovered by Zagier, by Almkvist and
Zudilin and by Cooper in their searches for integral solutions for certain
families of second- and third-order differential equations. We find new
representations, in terms of constant terms of powers of Laurent polynomials,
for all the 15 sporadic sequences.
  The new representations in turn lead to binomial expressions for the
sequences, which, as opposed to previous expressions, do not involve powers of
8 and powers of 3. We use these to establish the supercongruence $B_{np^k}
\equiv B_{np^{k-1}} \bmod p^{2k}$ for all primes $p \ge 3$ and integers $n,k
\ge 1$, where $B_n$ is a sequence discovered by Zagier and known as Sequence
$\mathbf{B}$.
  Additionally, for 14 out of the 15 sequences, the Newton polytopes of the
Laurent polynomials used in our representations contain the origin as their
only interior integral point. This property allows us to prove that these 14
sporadic sequences satisfy a strong form of the Lucas congruences, extending
work of Malik and Straub. Moreover, we obtain lower bounds on the $p$-adic
valuation of these 14 sequences via recent work of Delaygue.
