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Pascal's Triangle related question

  1. Aug 23, 2009 #1
    Let n and k be positive integers. After calculating several examples, guess a closed formula for:

    (n \ 0) + (n + 1 \ 1) + ... + (n + k \ k)

    If it helps, this is the formula for the sum of the nth row of the pascal triangle:

    (n \ 0) + (n \ 1) + .... (n \ k) = 2^n

    (n \ 0) means n choose 0. I couldn't write that in the forum so I had to improvise. Hopefully you know what it means. The "n" is on top and 0 is on the bottom.
     
  2. jcsd
  3. Aug 23, 2009 #2
    I know that answer, but I am foggy on how to get it.

    I will mention that

    [tex]\sum_{i=0}^{k} \left( _i^{n+i} \right) = \sum_{i=0}^{k} \left(_n^{n+i} \right)[/tex]

    by symmetry. I'm not sure that helps but it's a different angle to investigate.

    --Elucidus

    P.S. Look into d-simplex numbers too.
     
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