What is the asymptotic value of
$$\max\ \frac{(a+b)!(c+d)!b}{a!b!c!d!(b+d)}, \quad \textrm{if}\ \ a+b+c+d=n?$$
Is it $\approx 2^n/n$? Is it related to the asymptotic value of the Catalan numbers?
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1For some intuition, look at the easy case, where $a=b=c=d=n/4$. – Paul May 14 '15 at 18:51
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yup! that is the intution. In fact maximum value of binomial coefficient also indicates that. But is there any proof? – baban May 14 '15 at 18:54
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For @Paul's case, I get $2^n$, not $2^n/n$. – wythagoras May 14 '15 at 18:59
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may I have the deduction? – baban May 14 '15 at 19:08