14 Bertrand's postulate ●sums to (份=2” k=0 ·is symmetric: (风)=（n”) 1 331 From the functional equation ()=)one easily finds that for 4641 15101051 every n the binomial coefficients form a sequence that is symmetric 1 615201561 and unimodal:it increases towards the middle,so that the middle binomial 172135352171 coefficients are the largest ones in the sequence: Pascal's triangle 1=(8)<(份)<…<（)=(2)>…>(n)>(份）)=1. Here x resp.[r]denotes the number z rounded down resp.rounded up to the nearest integer. From the asymptotic formulas for the factorials mentioned above one can obtain very precise estimates for the sizes of binomial coefficients.How- ever,we will only need very weak and simple estimates in this book,such as the following:()<2m for all k,while for n>2 we have n 2n n/21 n with equality only for n =2.In particular,for n>1, 2n n 器 2 This holdssince(a middle binomial coefficient is the largest eny in the sequence ()+()(),(2),..(),whose sumis 2",and whose average is thus On the other hand,we note the upper bound for binomial coefficients 2 n(n-1)…(n-k+1) nk k kI which is a reasonably good estimate for the "small"binomial coefficients at the tails of the sequence,when n is large(compared to k). References [1]P.ERDOS:Beweis eines Satzes von Tschebyschef,Acta Sci.Math.(Szeged)5 (1930-32).194-198. [2]R.L.GRAHAM,D.E.KNUTH O.PATASHNIK:Concrete Mathematics. A Foundation for Computer Science,Addison-Wesley,Reading MA 1989. [3]G.H.HARDY E.M.WRIGHT:An Introduction to the Theory of Numbers, Fifth edition,Oxford University Press 1979. [4]P.RIBENBOIM:The New Book of Prime Number Records,Springer-Verlag, New York 1989.14 Bertrand’s postulate • sums to n k=0 n k  = 2n • is symmetric: n k  = n n−k  . From the functional equation n k  = n−k+1 k n k−1  one easily finds that for every n the binomial coefficients n k  form a sequence that is symmetric and unimodal: it increases towards the middle, so that the middle binomial 1 coefficients are the largest ones in the sequence: 1 1 1 1 1 1 1 1 2 3 4 5 6 7 15 10 6 3 1 1 4 10 20 35 15 5 1 1 6 7 1 21 2135 1 Pascal’s triangle 1 = n 0  < n 1  < ··· < n n/2  = n n/2  > ··· > n n−1  > n n  = 1. Here x resp. x denotes the number x rounded down resp. rounded up to the nearest integer. From the asymptotic formulas for the factorials mentioned above one can obtain very precise estimates for the sizes of binomial coefficients. How￾ever, we will only need very weak and simple estimates in this book, such as the following: n k  ≤ 2n for all k, while for n ≥ 2 we have  n n/2  ≥ 2n n , with equality only for n = 2. In particular, for n ≥ 1, 2n n  ≥ 4n 2n. This holds since n n/2  , a middle binomial coefficient, is the largest entry in the sequence n 0  + n n  , n 1  , n 2  ,..., n n−1  , whose sum is 2n, and whose average is thus 2n n . On the other hand, we note the upper bound for binomial coefficients  n k  = n(n − 1)···(n − k + 1) k! ≤ nk k! ≤ nk 2k−1 , which is a reasonably good estimate for the “small” binomial coefficients at the tails of the sequence, when n is large (compared to k). References [1] P. ERDOS˝ : Beweis eines Satzes von Tschebyschef, Acta Sci. Math. (Szeged) 5 (1930-32), 194-198. [2] R. L. GRAHAM, D. E. KNUTH & O. PATASHNIK: Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley, Reading MA 1989. [3] G. H. HARDY & E. M. WRIGHT: An Introduction to the Theory of Numbers, Fifth edition, Oxford University Press 1979. [4] P. RIBENBOIM: The New Book of Prime Number Records, Springer-Verlag, New York 1989