proposed
approved
editing
Cf. A031971, A185393, A249459, A349970.
a(n) = 2^n * A031971(n) =).
a(n) =
reviewed
a[n_] := Sum[If[k == n == 0, 1, (2*k)^n], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 07 2021 *)