Triangular powers of integers from determinants of binomial coefficient matrices

Abstract

Formulas are obtained for the determinants of certain matrices whose entries are zero and either binomial coefficients or their negatives. A consequence is that, for all integers n ≥ 2 and k ≥ 2, there exists an ( n − 1)( k − 1) × ( n − 1)( k − 1) matrix M ( n , k ) whose entries are the alternating binomial coefficients (−1) j +1 ( j n ) and zeros such that det( M ( n , k )) = ± k t n −1 , where t n −1 is the ( n − l)th triangular number. Further, if we form the infinite matrix whose k th row is ( 0 k ), ( 1 k ), ( 2 k ),…, then each of the above mentioned determinants is, up to sign, the determinant of an n × n submatrix A of obtained by selecting the initial n columns, and some choice of n rows of . The matrices M ( n , k ), and others that we will consider also have the unexpected property that det(| M ( n , k )|) = |det( M ( n , k )|, where | M | denotes the matrix obtained from M by replacing each entry with its absolute value.