In this paper, we derive relationships between weighted isobaric polynomials (WIP) and determinants of some Hessenberg matrices and inverse of some triangular matrices. Since weighted isobaric polynomials are a general form of several polynomials and number sequences which are linearly recurrent, any result obtained from these polynomials is applicable to in other areas. In this context, we give an application of our result on a new kind of Fibonacci numbers generalized in the distance sense.

integer sequences, weighted isobaric polynomials, triangular matrix, Hessenberg matrix.