Results 1 to 2 of 2

Thread: Combinatorics - identity: sum[0<=k<=n/2](-1)^k binom(n-k,k) 2^(n-2k) = n+1 ...?

  1. #1

    Question Combinatorics - identity: sum[0<=k<=n/2](-1)^k binom(n-k,k) 2^(n-2k) = n+1 ...?

    Maybe can someone help? Is the equation with n ≥ 0 correct? (easier to solve by using generating function)



    [tex]\displaystyle \mbox{2. }\quad \sum_{0\leq k \leq n/2}\, (-1)^k\, \binom{n\, -\, k}{k}\, 2^{n-2k}\, =\, n\, +\, 1\quad ?[/tex]

    [tex]\displaystyle \mbox{3. }\quad \sum_{k=0}^{n}\, \binom{n\, +\, k}{2k}\, 2^{n-k}\, =\, \dfrac{2^{2n+1}\, +\, 1}{3}\quad ?[/tex]

    [tex]\displaystyle \mbox{4. }\quad \sum_{k=0}^n\, \binom{2k}{k}\, \binom{n}{k}\, \left(-\dfrac{1}{4}\right)^k\, =\, 2^{-2n}\, \binom{2n}{n}\quad ?[/tex]

    [tex]\displaystyle \mbox{5. }\quad \sum_k\, \binom{m}{k}\, \binom{n\, +\, k}{m}\, =\, \sum_k\, \binom{m}{k}\, \binom{n}{k}\, 2^k\quad ?[/tex]

    . . . . .[tex]\mbox{Here }\, \binom{n}{k}\, = 0\, \mbox{ if }\, k\, >\, n.[/tex]
    Attached Images Attached Images
    Last edited by stapel; 01-11-2018 at 05:20 PM. Reason: Typing out the text in the graphic; creating useful subject line.

  2. #2
    Elite Member stapel's Avatar
    Join Date
    Feb 2004
    Posts
    15,851

    Cool

    Quote Originally Posted by evebart View Post
    Maybe can someone help? Is the equation with n ≥ 0 correct? (easier to solve by using generating function)



    [tex]\displaystyle \mbox{2. }\quad \sum_{0\leq k \leq n/2}\, (-1)^k\, \binom{n\, -\, k}{k}\, 2^{n-2k}\, =\, n\, +\, 1\quad ?[/tex]

    [tex]\displaystyle \mbox{3. }\quad \sum_{k=0}^{n}\, \binom{n\, +\, k}{2k}\, 2^{n-k}\, =\, \dfrac{2^{2n+1}\, +\, 1}{3}\quad ?[/tex]

    [tex]\displaystyle \mbox{4. }\quad \sum_{k=0}^n\, \binom{2k}{k}\, \binom{n}{k}\, \left(-\dfrac{1}{4}\right)^k\, =\, 2^{-2n}\, \binom{2n}{n}\quad ?[/tex]

    [tex]\displaystyle \mbox{5. }\quad \sum_k\, \binom{m}{k}\, \binom{n\, +\, k}{m}\, =\, \sum_k\, \binom{m}{k}\, \binom{n}{k}\, 2^k\quad ?[/tex]

    . . . . .[tex]\mbox{Here }\, \binom{n}{k}\, = 0\, \mbox{ if }\, k\, >\, n.[/tex]
    We'll be glad to help, but first we'll need to know where you're having trouble. What methods are you expected to use? What have you tried? How far have you gotten?

    Please be complete. Thank you!

Bookmarks

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •