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On sums of binomial coefficients and their applications  ZhiWei Sun
;  Date: 
21 Apr 2004  Subject:  Number Theory; Combinatorics MSCclass: 11B65; 05A19; 11B37; 11B68  math.NT math.CO  Abstract:  In this paper we study recurrences concerning the combinatorial sum $S(n,r)=sum_{kequiv r (mod m)}inom {n}{k}$ and the alternate sum $sum_{kequiv r (mod m)}(1)^{(kr)/m}inom{n}{k}$, where $m>0$, $nge 0$ and $r$ are integers. For example, we show that if $nge m1$ then $$sum_{i=0}^{lfloor(m1)/2
floor}(1)^iinom{m1i}i S(n2i,ri)=2^{nm+1}.$$ We also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity given by the author [Integers 2(2002)].  Source:  arXiv, math.NT/0404385  Services:  Forum  Review  PDF  Favorites 


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