Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove ∑ … = (2^n-1)n!

  1. Feb 19, 2009 #1

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Prove, for any integer n:

    [tex]\sum_{0\,\leq\,m\,< n/2}\,(-1)^m(n - 2m)^n\,^nC_m\ =\ 2^{n-1}\,n![/tex]

    for example, 77 - 577 + 3721 - 1735

    = 823543 - 546875 + 45927 - 35 = 304560 = 64 times 5040
     
  2. jcsd
  3. Feb 19, 2009 #2

    Gokul43201

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I'm guessing there is a brute force way to prove this as well as a clever combinatoric argument, and you're hoping we only find the first, so you can dazzle us with the second? :biggrin:
     
  4. Feb 19, 2009 #3

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Gokul! :smile:

    Sort of … I accidentally found a geometric-cum-combinatoric proof of this while looking at a homework thread,

    but I couldn't help thinking that there must be some way of solving this just by looking at it and coming up with a solution …

    but no ordinary technique comes to mind since the exponand (is that the right word? :redface:) keeps changing.

    I was hoping somebody knew a finding-the-solution technique (maybe for a simpler problem), rather than an already-knowing-what the-solution-is technique! :smile:
     
  5. Mar 24, 2009 #4
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Prove ∑ … = (2^n-1)n!
  1. Proving 2=1 (Replies: 11)

  2. Traveling 'n such (Replies: 15)

  3. Prove 1+1 = 2 (Replies: 56)

Loading...