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

翻译：本文介绍并研究了一种新的整数序列，称为高阶梅森序列。该序列类似于高阶斐波那契数，且与梅森数密切相关。我们讨论了该新序列的多种代数性质，如比内公式、卡塔兰恒等式、德奥卡涅恒等式、生成函数、有限和与二项式和等，以及其与梅森数和雅可比数的相互关系。此外，我们研究了由高阶梅森数的二项式变换生成的序列，给出了它们的递推关系和代数性质。最后，我们给出了高阶梅森数的矩阵生成元及三对角矩阵表示。