The number of “magic” squares and hypercubes

We define a semi-magic square to be a square matrix whose entries are nonnegative integers and whose rows and columns (that is, lines) sum up to the same number. A magic square is a semimagic square whose main diagonals also add up to the line sum. A symmetric magic square is a magic square which is a symmetric matrix. A pandiagonal magic square is a semi-magic square whose diagonals parallel to the main diagonal from the upper left to the lower right, wrapped around, add up to the line sum. These definitions clash somewhat with that of the classical magic square, a magic square of order n whose entries are the integers 1, . . . , n2.