  Let $a$ and $b$ be positive integers with prime factorisations $a =
p_1^np_2^n$ and $b = q_1^nq_2^n$. We prove that the number of essentially
distinct $\alpha$-graceful labelings of the complete bipartite graph $K_{a, b}$
equals the alternating sum of fourth powers of binomial coefficients
$(-1)^n[\binom{2n}{0}^4 - \binom{2n}{1}^4 + \binom{2n}{2}^4 - \binom{2n}{3}^4 +
\cdots + \binom{2n}{2n}^4]$.
