ISL 1979 Problem 19

For $k = 1, 2, \ldots$ consider the $k$ -tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that $$a_1 + 2a_2 + \cdots + ka_k = 1979.$$ Show that there are as many such $k$ -tuples with odd $k$ as there are with even $k$ .