Pascal-like matrix with double factorial binomial coefficients

Key Points

• The research aims to explore a new variation of a Pascal-like matrix using double factorial binomial coefficients and its properties.
• Defined a new Pascal-like matrix with double factorial binomial coefficients.
• Calculated the inverse of the proposed matrix.
• Derived explicit representations for the matrix's powers under various exponent types (integer, rational, irrational).
• Generalized the matrix and established new identities.
• Demonstrated key properties of the new matrix variant.
• Successfully calculated the inverse of the matrix.
• Obtained explicit forms for powers of the matrix for different types of exponents.
• Presented several new mathematical identities related to the generalized matrix.