Abstract
For a Riordan array \(R=(r_{n,k})\), we call the matrix with entries \( r^{(m)}_{n,k} =\sum _{j=k}^{n} r^{(m-1)}_{n,j} \) the mth Bernoulli triangle of R for \(m\ge 2\), with \( r^{(1)}_{n,k} = r_{n,k}\). In this paper, using Riordan array method, we investigate sums along straight lines of indices in Bernoulli triangles of Pascal, Catalan, and Delannoy matrices, and obtain some identities involving the binomial coefficients, the generalized Fibonacci numbers, and some other combinatorial numbers.
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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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The authors thank the referees and editor for their valuable suggestions which improved the quality of this paper.
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This work is supported by the National Natural Science Foundation of China (Grant No.11861045, 12101280).
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Yang, L., Yang, SL. & He, TX. Summations on the Diagonals of a Riordan Array and Some Applications. Graphs and Combinatorics 39, 76 (2023). https://doi.org/10.1007/s00373-023-02676-2
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DOI: https://doi.org/10.1007/s00373-023-02676-2