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Unimodal sequence - proof for binomial coeff

  1. Jan 6, 2009 #1
    Hi
    question in book states:
    show that the sequence <a_0,a_1,a_2,...,a_n> where a_r = (C(n,r))(x^(n-r))(y^r) = F(n,r), where
    x and y are positve, is unimodal
    unimodal means there exists a positve integer 1 < j < n such that
    a_1 < a_2 < ...... < a_(j -1) <= a_j > a_(j+1) > .... a_n
    note also that C is n!/((n-r)!)r!

    books solution says determine the sequence is strictly increasing for

    r < t = (ny-x)/(x+y) -------EQUATION 1

    and strictly decreasing for r > t. there are 4 possibilites

    (1) t < 0, then a_0 > a_1 > a_2 > ... > a_n
    (2) t = 0, then a_0=a_1 > a_2 > ... > a_n
    (3) t > 0 , and t is not an integer. let k = floor(t) +1 ; then
    a_1 < a_2 < ...... < a_k > a_(k+1) > .... > a_n
    (4) t > 0 and t is integer. then
    a_1 < a_2 < ...... < a_t = a_(t+1) > a_(t+2) > .... > a_n


    i do not get EQUATION 1, but i can get

    r < t = (ny-(1/x))/((1/x)+y) -------EQUATION 2

    by dividing F(n,r+1) / F(n,r)

    if EQUATION 2 is correct then there is no way (1) and (2) are valid ?

    anyone agree with me or am i talking gibberish :)
     
  2. jcsd
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