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Combinatorial and Additive Number Theory pp 257–312Cite as

On Sums Related to Central Binomial and Trinomial Coefficients

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Abstract

A generalized central trinomial coefficient T n (b, c) is the coefficient of x n in the expansion of \((x^{2} + bx + c)^{n}\) with \(b,c \in \mathbb{Z}\). In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic forms, and 62 proposed new series for 1∕π motivated by congruences and related dualities.

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Acknowledgements

The work was supported by the National Natural Science Foundation (grant 11171140) of China, and the initial version of this paper was posted to arXiv in Jan. 2011 as a preprint with the ID arXiv:1101.0600. The preprint version of this paper available from arXiv has stimulated some others to work on our conjectural series for 1∕π of types I–V in Sect. 5.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Zhi-Wei Sun

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  1. Zhi-Wei Sun
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Correspondence to Zhi-Wei Sun .

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Editors and Affiliations

  1. Department of Mathematics, Lehman College (CUNY), Bronx, New York, USA

    Melvyn B. Nathanson

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Sun, ZW. (2014). On Sums Related to Central Binomial and Trinomial Coefficients. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, vol 101. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1601-6_18

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