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Stirling's approximation isn't needed here View a pdf of the paper titled stirling's approximation for central extended binomial coefficients, by steffen eger The approximation and its equivalent form can be obtained by rearranging stirling's extended formula and observing a coincidence between the resultant power series and the taylor series expansion of the hyperbolic sine function.
We have done so by using simple probability theory and information theory. \) entropy bound for binomial coefficients This behavior is captured in the approximation known as stirling's formula ((also known as stirling's approximation))
Stirling's formula the factorial function n
∼ 2 π n (n e) n n Furthermore, for any positive integer n n, we have the bounds 2 π n n + 1 2 e − n ≤ n ≤ e n n + 1 2 e − n In confronting statistical problems we often encounter factorials of very large numbers.
Ion formulas for central extended binomial coefficients is as follows First, we determine the asymptotic distribution of the sum of discrete uniform variables, which we easi Apply the above stirling’s approximation, we have Upper/lower bound for binomial coefficients
\ ((n k)k ≤ (n k) ≤ (en k)k
(12) (12) (n k) k ≤ (n k) ≤ (e n k) k
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