A congruence related to central trinomial coefficients

ZHANG Wenbin; ZHANG Yong; WU Jiachen

doi:10.13471/j.cnki.acta.snus.ZR20240034

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A congruence related to central trinomial coefficients

Research articles | 更新时间：2025-11-14

- A congruence related to central trinomial coefficients
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 64, Issue 4, Pages: 102-108(2025)
- 作者机构：
  
  1.广州华商学院人工智能学院，广东广州 511300
  2.南京工程学院数理学院，江苏南京 211167
- 作者简介：
  
  ZHANG Wenbin（wenbinzhang1983@163.com）
  ZHANG Yong（yongzhang1982@163.com）
  WU Jiachen（wujiachen2024@163.com）
- 基金信息：
  
  南京工程学院数理学院, 江苏南京(11971222;12071208)
- DOI：10.13471/j.cnki.acta.snus.ZR20240034
  CLC： O156.1
- Received：26 January 2024，
  
  Accepted：18 November 2024，
  
  Published Online：21 April 2025，
  
  Published：25 July 2025
- 稿件说明：
移动端阅览
.一个与中心三项式系数有关的同余式[J].中山大学学报(自然科学版)(中英文),2025,64(04):102-108.

ZHANG Wenbin,ZHANG Yong,WU Jiachen.A congruence related to central trinomial coefficients[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(04):102-108.
.一个与中心三项式系数有关的同余式[J].中山大学学报(自然科学版)(中英文),2025,64(04):102-108. DOI： 10.13471/j.cnki.acta.snus.ZR20240034.

ZHANG Wenbin,ZHANG Yong,WU Jiachen.A congruence related to central trinomial coefficients[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(04):102-108. DOI： 10.13471/j.cnki.acta.snus.ZR20240034.

摘要

定义中心三项式系数

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832579&type=

3.72533321

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832582&type=

3.21733332

为

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832585&type=

4.91066647

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832633&type=

19.38866615

展开式中

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832637&type=

2.96333337

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832616&type=

2.79399991

项的系数. 我们证明了孙智伟提出的关于中心三项式系数与中心二项式系数和式的同余式猜想.

Abstract

The central trinomial coefficient

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832504&type=

3.72533321

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832506&type=

3.21733332

denotes the coefficient of

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832547&type=

2.96333337

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832559&type=

2.79399991

in the expansion of

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832562&type=

4.91066647

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84832555&type=

19.38866615

. We prove a congruence related to the sums of the central trinomial coefficient and the central binomial coefficient

which was conjectured by Z.-W. Sun.

关键词

Keywords

references

BEUKERS F ， 1985 . Some congruences for the Apéry numbers ［J］. J Number Theory ， 21 （ 2 ）： 141 - 155 .

GESSEL I M ， 1983 . Some congruences for generalized Euler numbers ［J］. Can J Math ， 35 （ 4 ）： 687 - 709 .

GOULD H W ， 1972 . Combinatorial identities ［M］. West Virginia ： Morgantown Printing and Binding Co .

GRANVILLE A ， 1997 . Arithmetic properties of binomial coefficients I： Binomial coefficients modulo prime powers ［C］// Canadian Mathematical Society Conference Proceedings ， 20 ： 253 - 275 .

OSBURN R ， SAHU B ， STRAUB A ， 2016 . Supercongruences for sporadic sequences ［J］. Proc Edinb Math Soc ， 59 （ 2 ）： 503 - 518 .

SUN Z W ， 2014 . On sums related to central binomial and trinomial coefficients ［M］// NATHANSON M. Combinatorial and additive number theory： CANT2011 and 2012 . New York ： Springer .

SUN Z W ， 2019 . Open conjectures on congruences ［J］. Nanjing Univ J Math Biquarterly ， 36 （ 1 ）： 1 - 99 .

WANG C ， SUN Z W ， 2024 . Some parametric congruences involving generalized central trinomial coefﬁcients ［J］. Rev Real Acad Cienc Exactas Fís Nat Ser A-Mat ， 118 ： 3 .

ZHANG Y ， 2021 . Three supercongruences involving generalized central trinomial coefficients ［J］. Adv Math（China）， 50 （ 2 ）： 184 - 194 .

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ZHANG Wenbin

ZHANG Yong

WU Jiachen

Related Institution

Institute of Artificial Intelligence, Guangzhou Huashang College

School of Mathematics and Physics, Nanjing Institute of Technology

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