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In this article, we introduce and study a new integer sequence referred to asthe higher order Mersenne sequence. The proposed sequence is analogous to thehigher order Fibonacci numbers and closely associated with the Mersennenumbers. Here, we discuss various algebraic properties such as Binet's formula,Catalan's identity, d'Ocagne's identity, generating functions, finite andbinomial sums, etc. of this new sequence, and some inter-relations withMersenne and Jacobsthal numbers. Moreover, we study the sequence generated fromthe binomial transforms of the higher order Mersenne numbers and present therecurrence relation and algebraic properties of them. Lastly, we give matrixgenerators and tridiagonal matrix representation for higher order Mersennenumbers.

The sequence of higher order Mersenne numbers and associated binomial transforms