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Let $n\in\mathbb{N}$. For $0\le l\le n$ consider \begin{equation} b_l:=4^{-l} \sum_{j=0}^l \frac{\binom{2 l}{2 j} \binom{n}{j}^2}{\binom{2 n}{2 j}}\text{.} \end{equation} Do you know a technique how to prove that \begin{equation} b_l\ge b_n\text{,$\quad 0\le l\le n-1$?} \end{equation} Going through a long list of binomial identities I did not find epiphany.

Addition: Plot of $b_l$ for $n=20$. Plot of $b_l$ for $n=20$

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    May be $b_l$ can be interpreted as probability of some event.2012-08-26
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    precarious: Why/How do you know the result holds?2012-08-26
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    @did Numerical experiments gave sufficient evidence.2012-08-26
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    @precarious I have a solution, I hope it is correct.2012-08-26
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    @precarious before giving any answer do you have any simulation result for $l=10^6,...,10^7$?2012-08-26
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    precarious: I am not sure you answered my question.2012-08-26
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    @SeyhmusGüngören: I do not have it yet. Maxima is calculating. I guess that Mathematica would be much faster.2012-08-26
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    @did: Could you please be more precise?2012-08-26
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    precarious: Well, you certainly did not wake up one morning saying to yourself: "Hey, today let me prove that [complicated sum of fractions of products of binomial coefficients] is greater than [other complicated sum of fractions of products of binomial coefficients]"... There is a reason why you are interested in these sums and there is a reason why you believe they behave the way you ask us to prove they behave--but, sadly, you mentioned none.2012-08-26
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    @did: I think that your assessment is unfair. Of course, there is a reason why I am interested in the problem. Your original question, however, is different and completely answered above.2012-08-26
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    *I think that your assessment is unfair*. OK. You might mention which part(s) is (are) unfair and why. *Of course, there is a reason why I am interested in the problem*. OK. You might wish to mention this reason. If you do not want to, just say so. *Your original question, however, is different and completely answered above*. Certainly not. (And let us leave it at that, shall we.)2012-08-26

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The sum $b_\ell$ is clearly hypergeometric: $$ b_\ell = 4^{-\ell} \sum_{j=0}^\ell \frac{\binom{2\ell}{2 j} \binom{n}{j}^2}{\binom{2n}{2j}} = 4^{-\ell} \sum_{j=0}^\ell \frac{(-n)_j}{\left(\frac{1}{2}-n\right)_j} \frac{(-\ell)_j \left(\frac{1}{2}-\ell\right)_j}{j! \cdot j!} = 4^{-\ell} {}_3F_2\left(\left. \begin{array}{ccc} -\ell & \frac{1}{2} -\ell & -n \\ & 1 & \frac{1}{2}-n \end{array} \right| 1\right) $$ This representation allows to find $$ b_n = 4^{-n} {}_3F_2\left(\left. \begin{array}{ccc} -n & \frac{1}{2} -n & -n \\ & 1 & \frac{1}{2}-n \end{array} \right| 1\right) = 4^{-n} {}_2F_1\left(\left. \begin{array}{cc} -n & -n \\ & 1 \end{array} \right| 1\right) = \frac{1}{4^n} \binom{2n}{n} $$

The above function allows to extend the sequence to $\ell > n$. This sequence is not decreasing for all $\ell \geqslant 0$, but does appear to decrease on the interval $0 \leqslant \ell\leqslant n$. Here is an example for $n=20$: enter image description here


Now, to the probabilistic interpretation of the $b_\ell$. Suppose an urn contains $2n$ balls, $n$ white and $n$ blue. We sample $m$ balls without the replacement. The probability that the sample contains equal number of balls of different colors is $$ p_m = \cases{\frac{\binom{n}{j} \binom{n}{j}}{\binom{2n}{2j}} & $m=2j$ \\ 0 & $m = 2j+1$} $$ If the size of the sample follows a symmetric binomial distribution, the probability of getting sample with equal number of colors is: $$ b_\ell = \sum_{j=0}^\ell \frac{\binom{n}{j} \binom{n}{j}}{\binom{2n}{2j}} \binom{2\ell}{2j} 4^{\ell} $$

I am not seeing how to establish the inequality though, but hope this helps.

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    do you have results for $l=10^6,...,10^7$?2012-08-26
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    @SeyhmusGüngören What results are you interested in?2012-08-26
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    I am interested in $b_l$ as a function of $l$ when $l$ is big enough such as in the range $10^6$ to $10^7$2012-08-26
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    But $b_\ell$ is also a function of $n$. If you have access to _Mathematica_, use the expression in terms of the hypergeometric function I provided.2012-08-26
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    I dont have it right now. My Matlab is poor about such kinda things. let $n$ be $10^7$ and $l\in[10^6,10^7]$ could you please? if you nevermind?2012-08-26
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    @Sasha: Thank you for your answer. Your hypergeometric representation is interesting. Even though it does not yield an immediate solution it might be helpful.2012-08-26
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    @precarious I have some doubts that your function satisfies your conditions when $l$ is large. Meanwhile $n>l$ right?2012-08-26
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    @SeyhmusGüngören $l$ is considered to be in the range $0..n$.2012-08-26
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    @precarious ok if $l$ is in the range $0,...,n$ then $n>l$ as I said. What is the sense to have some values when $n$b_\ell = \sum_{j=0}^\ell A \binom{2\ell}{2j} 4^{\ell}$ is not always decreasing in $l$ especially when $l$ is quite large. Thats why I wanted to see the function for large $l$2012-08-26
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    @SeyhmusGüngören As far as I can see it, Sasha plotted values for $n$b_l$ can be extended in a natural way. I do not see an immediate impact concerning the problem. In your formula, what is $A$? – 2012-08-26
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    @precarious I took it as a constant to simplify the things. In your formula of course it does not but that part is not dependent on $l$.2012-08-26
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    @SeyhmusGüngören I am not sure whether such a simplification is admissible. So far, I did not find a counterexample. Computing $b_l$ for large $n$ and $l$ is quite expensive.2012-08-26