I would like to see if $ b_l:=4^{-l} \sum_{j=0}^l \frac{\binom{2 l}{2 j} \binom{n}{j}^2}{\binom{2 n}{2 j}}\text{.} $ is decreasing when $l$ is large enough say around $10^6$. I dont need any theoretical derivations though I wrote the following code part
b=zeros(1,10^7-10^6); for l=10^6:10^7-1 for j=0:l b(l-999999) = b(l-999999) + (nchoosek(2*l,2*j)*nchoosek(10^7,j)^2) / (nchoosek(2*10^7,2*j)); end end
EDIT:
This question is a simplified version of the original one. I intended to carry the matter here since what I found might imply that the function is no more decreasing for larger $l$. Please see the discussion over there.
Inequality involving sums of fractions of products of binomial coefficients
I couldnt make use of Stirlings approximation. Here is the changed code part:
b=zeros(1,10^7-10^6);
for l=10^6:10^7-1
for j=0:l b(l-999999) = b(l-999999) + ((sqrt(2*pi*2*l)*(2*l/exp(1))^(2*l))/((sqrt(2*pi*2*j)*(2*j/exp(1))^(2*j)*((sqrt(2*pi*2*l)*(2*l/exp(1))^(2*l))-(sqrt(2*pi*2*j)*(2*j/exp(1))^(2*j)))))*... ((sqrt(2*pi*10^7)*(10^7/exp(1))^(10^7))/((sqrt(2*pi*j)*(j/exp(1))^(j)*((sqrt(2*pi*10^7)*(10^7/exp(1))^(10^7))-(sqrt(2*pi*j)*(j/exp(1))^(j))))))/... (sqrt(2*pi*2*10^7)*(2*10^7/exp(1))^(2*10^7))/((sqrt(2*pi*2*j)*(2*j/exp(1))^(2*j)*((sqrt(2*pi*2*10^7)*(2*10^7/exp(1))^(2*10^7))-(sqrt(2*pi*2*j)*(2*j/exp(1))^(2*j)))))); end
end
which cannot provide me any result due to the accuracy of nchoosek
, i.e., big numbers are creating problems. Do you have any idea how I can deal with this problem? I only want to know if the function is decreasing or not.
Any help will be appreciated.
Thanks in advance.