A New Triangle Generation of Some Generalized Genocchi Numbers

The two-dimensional rook theory can be generalized to three and higher dimensions by assuming that rooks attack along hyperplanes. Using this generalization, Alayont and Krzywonos defined two separate families of boards in three and higher dimensions generalizing the two-dimensional triangular boards whose rook numbers correspond to generalizations of Stirling numbers of the second kind and Genocchi numbers. This combinatorial interpretation of the Genocchi numbers was shown to provide a new triangle generation of the Genocchi numbers. In this paper, we prove similar triangle generations for the third and fourth generalized Genocchi numbers using rook numbers of boards in four and five dimensions.