Congruences involving generalized central trinomial coefficients  

作　　者：SUN Zhi-Wei 

机构地区：[1]Department of Mathematics,Nanjing University

出　　处：《Science China Mathematics》2014年第7期1375-1400,共26页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.11171140);the PAPD of Jiangsu Higher Education Institutions

摘　　要：For integers b and c the generalized central trinomial coefficient Tn(b,c)denotes the coefficient of xnin the expansion of(x2+bx+c)n.Those Tn=Tn(1,1)(n=0,1,2,...)are the usual central trinomial coefficients,and Tn(3,2)coincides with the Delannoy number Dn=n k=0n k n+k k in combinatorics.We investigate congruences involving generalized central trinomial coefficients systematically.Here are some typical results:For each n=1,2,3,...,we have n-1k=0(2k+1)Tk(b,c)2(b2-4c)n-1-k≡0(mod n2)and in particular n2|n-1k=0(2k+1)D2k;if p is an odd prime then p-1k=0T2k≡-1p(mod p)and p-1k=0D2k≡2p(mod p),where(-)denotes the Legendre symbol.We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.For integers b and c the generalized central trinomial coefficient Tn（b,c） denotes the coefficient of xnin the expansion of（x2＋ bx ＋ c）n.Those Tn = Tn（1,1）（n = 0,1,2,...） are the usual central trinomial coefficients,and Tn（3,2） coincides with the Delannoy number Dn = n k=0 n k n＋k k in combinatorics.We investigate congruences involving generalized central trinomial coefficients systematically.Here are some typical results： For each n = 1,2,3,...,we have n-1 k=0（2k ＋ 1）Tk（b,c）2（b2- 4c）n-1-k≡ 0（mod n2）and in particular n2｜ n-1k=0（2k ＋ 1）D2k; if p is an odd prime then p-1 k=0T2k≡-1p（mod p） and p-1 k=0D2k≡ 2p（mod p）,where（-） denotes the Legendre symbol.We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.

关 键 词：CONGRUENCES central trinomial coefficients Motzkin numbers central Delannoy numbers 

分 类 号：O174.14[理学—数学]