Stanley, building on work of Stern, defined an array of numbers as follows: the first row of the array is ···0001000···. Each successive row is obtained by copying down the previous row and inserting, between each pair of numbers from the previous row, the sum of that pair of numbers. Let s_n^r be the sum of the r-th powers of the elements in the n-th row of the array. Stanley showed that, for each positive integer r, the sequence s_n^r obeys a homogeneous linear recurrence in n of length r/2 + O(1). Numerical evidence, however, suggested that s_n^r obeys shorter recurrences, of length r/3 + O(1). We prove Stanley's conjecture.