BACKGROUND Reporting of absolute risk difference (RD) is recommended for clinical and epidemiological prospective studies. In analyses of multicenter studies, adjustment for center is necessary when randomization is stratified by center or when there is large variation in patients outcomes across centers. While regression methods are used to estimate RD adjusted for baseline predictors and clus...

We consider the distribution defined by the reciprocals of binomial coefficients and compute the corresponding moments. We find recurrence relations and the relative ordinary generating functions, which we give explicitly for the first six moments (m = 0, 1, . . . , 5). Finally we give asymptotic approximations of the moments and of related quantities.

In 1913 A. Fleck proved that if p is a prime, and n > 0 and r are integers then ∑ k≡r (mod p) (n k ) (−1) ≡ 0 ( mod pb(n−1)/(p−1)c ) . Only recently the significance of Fleck’s congruence was realized. It plays a fundamental role in Colmez’ and Wan’s investigation of the ψ-operator related to Fontaine’s theory and p-adic Langlands correspondence. In this talk we give a survey of the recent deve...

In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let p be a prime and let a be any positive integer. We determine ∑pa−1 k=0 ( 2k k+d ) mod p2 for d = 0, . . . , pa and ∑pa−1 k=0 ( 2k k+δ ) mod p3 for δ = 0, 1. We also show that

In this paper we consider the generalized Catalan numbers F (s, n) = 1 (s−1)n+1 ( sn n ) , which we call s-Catalan numbers. For p prime, we find all positive integers n such that p divides F (p, n), and also determine all distinct residues of F (p, n) (mod p), q ≥ 1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second...

so that there are infinitely many numbers occurring at least 4 times in the Pascal Triangle. Stripped of these trivialities, the more interesting problem becomes to determine the natural numbers that occur at least twice as binomial coefficients of the shape ( n k) with 2 k 1 2n, and this is yet unsolved in its full generality. The only nontrivial solutions known at this time are the following:...

Recommended by George Andrews We investigate the integral representation of infinite sums involving the reciprocals of triple bino-mial coefficients. We also recover some wellknown properties of ζ3 and extend the range of results given by other authors.

Assuming the abc-conjecture we show that there are only nitely many powerful binomial coeecients ? n k with 3 k n=2; in fact, that if q 2 divides ? n k that q n 2 ? n k o(1). Unconditionally we show that there are N 1=2+o(1) powerful binomial coeecients in the top N rows of Pascal's Triangle.