Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows easily from Cauchy-Binet identity.) Later Turán published a simpler proof for the sum of the fourth powers but in Chinese. I vaguely remember that there are simpler probabilistic proofs for both cases.

My question is about simple proofs for these identities, especially the one for 4th powers.

Is there a formula for the 6th power?