CTcong.txt
swMATH ID:  30553 
Software Authors:  Chen, William Y. C.; Hou, QingHu; Zeilberger, Doron 
Description:  Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences. Many combinatorial sequences (e.g. the Catalan and the Motzkin numbers) may be expressed as the constant term of (P(x)^kQ(x)), for some Laurent polynomials (P(x)) and (Q(x)) in the variable (x) with integer coefficients. Denoting such a sequence by (a_k), we obtain a general formula that determines the congruence class, modulo (p), of the indefinite sum (sumlimits_{k=0}^{rp1} a_k), for any prime (p), and any positive integer (r), as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our computers) to automatically discover and prove congruence theorems for such partial sums. Moreover, we show that in many cases, the set of the residues is finite, regardless of the prime (p). 
Homepage:  http://sites.math.rutgers.edu/~zeilberg/tokhniot/CTcong.txt 
Dependencies:  Maple 
Keywords:  polynomials; combinatorial sequences; Legendre symbol; Laurent series 
Related Software:  MultiZeilberger; qTSPP; ContMarkovWZ; MarkovAZ; MarkovWZ; OEIS 
Cited in:  5 Publications 
Standard Articles
1 Publication describing the Software, including 1 Publication in zbMATH  Year 

Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences. Zbl 1368.11020 Chen, William Y. C.; Hou, QingHu; Zeilberger, Doron 
2016

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top 5
Cited by 6 Authors
2  Apagodu, Moa 
2  Hou, Qinghu 
2  Zeilberger, Doron 
1  Chen, Shaoshi 
1  Chen, William YongChuan 
1  Wang, Yushan 
Cited in 3 Serials
2  International Journal of Number Theory 
1  American Mathematical Monthly 
1  Journal of Difference Equations and Applications 
Cited in 3 Fields
4  Combinatorics (05XX) 
4  Number theory (11XX) 
1  Computer science (68XX) 