Sum of Terms of Magic Cube/Sequence

Sequence of Sums of Terms of Magic Cubes

The sequence of the sum totals of all the entries in magic cubes of order $n$ begins:

$1, \paren {36,} \, 378, 2080, 7875, 23 \, 436, 58 \, 996, 131 \, 328, \ldots$

However, note that while $36 = \dfrac {2^3 \paren {2^3 + 1} } 2$, a magic cube of order $2$ does not actually exist.

This sequence is A037270 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).