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Topic: Sum of squares of binomial coefficients
Replies: 5   Last Post: Oct 18, 2012 4:28 AM

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 Gavin Wraith Posts: 20 Registered: 9/29/06
Re: Sum of squares of binomial coefficients
Posted: Oct 11, 2012 5:55 PM

In message <111020121212107442%edgar@math.ohio-state.edu.invalid>
Jérôme Collet <Jerome.Collet@laposte.net> wrote:

> I need to compute the sum :
> \sum_{r,s}{ (\binom{r+s}{r} \binom{2m-r-s}{m-r})^2  }
> I know, because I used Stirling formula, Taylor-polynomials, and
> ignored some problems on the borders, that this sum should be close to
> \sqrt{2\pi m}.
> The convergence is very fast, error is less than .5% if m>7.
> Nevertheless, I do not know how to prove it correctly.

This statement worries me. The expression you give
\sum_{r,s}{ (\binom{r+s}{r} \binom{2m-r-s}{m-r})^2  }
is certainly larger than
\sum_{r,s}{ (\binom{r+s}{r} \binom{2m-r-s}{m-r})  }
which evaluates to \binom{2m}{m}, unless I am mistaken.
And that grows much faster than \sqrt{2\pi m}.

--
Gavin Wraith (gavin@wra1th.plus.com)