On members of Lucas sequences which are either products of factorials or product of middle binomial coefficients and Catalan numbers

Abstract

Let \(\{U_n\}_{n\ge 0}\) be a Lucas sequence. We show that if \(U_n\) is a product of factorials, then \(n\in \{1,2, 3, 4, 6, 8, 12\}\). Further we show that if \(U_n\) is a product of Catalan numbers or the middle binomial coefficients, then \(n\in \{1,2, 3, 4, 6, 8, 12, 16\}\). Here the \(m-\)th middle binomial coefficient is \(B_m=\left( {\begin{array}{c}2m\\ m\end{array}}\right) \) and the \(m-\)th Catalan number is \(C_m=\frac{1}{m+1}\left( {\begin{array}{c}2m\\ m\end{array}}\right) \).

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References

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    Laishram, S., Luca, F., Sias, M.: On members of Lucas sequences which are products of Catalan numbers. Int. J. Number Theory. (2021). https://doi.org/10.1142/S1793042121500457

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Acknowledgements

The author acknowledges the support of SERB MATRICS Project. The author would like to thank the referee for the careful reading of the manuscript.

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Correspondence to Shanta Laishram.

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Laishram, S. On members of Lucas sequences which are either products of factorials or product of middle binomial coefficients and Catalan numbers. Res. number theory 7, 29 (2021). https://doi.org/10.1007/s40993-021-00257-x

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