Let ${alpha}$ = (${alpha}_1,;{alpha}_2$,...) and ${eta}$ = (${eta}_1,;{eta}_2$,...) be two sequences with ${alpha}_1$ = ${eta}_1$ and k and n be natural numbers. We denote by $A^{(k,{pm})}_{{alpha},{eta}}(n)$ the matrix of order n with coefficients ${alpha}_{i,j}$ by setting ${alpha}_{1,i}$ = ${alpha}_i,;{alpha}_{i,1}$ = ${eta}_i$ for 1 ${leq}$ i ${leq}$ n and $${alpha}_{i,j}={{alpha}_{i-1,j-1}+{alpha}_{i-1,j};if;j{equiv}$$2,3,4,..., k + 1 (mod 2k) $${{alpha}_{i-1,j-1}-{alpha}_{i-1,j};if;j{equiv}$$ k + 2,..., 2k + 1 (mod 2k) for 2 ${leq}$ i, j ${leq}$ n. The aim of this paper is to study the determinants of such matrices related to certain sequence ${alpha}$ and ${eta}$ and some natural numbers k.