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pouzdrotf 2022-07-08 Answered
How to show that m = 1 n / 4 ( n m ) 2 n exp ( n / 8 )
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Ordettyreomqu
Answered 2022-07-09 Author has 22 answers
For convenience, lets replace n 4 n, so that it is equivalent to show m = 1 n ( 4 n m ) 2 4 n e n / 2 . We need a good bound for the partial sum in LHS, so consider
(for  m < n ) ( 4 n m 1 ) = ( 4 n ) ! m ! ( 4 n m ) ! m 4 n m + 1 < ( 4 n m ) 1 3
As 1 + 1 3 + 1 3 2 + = 3 2 , we get the bound m = 1 n ( 4 n m ) < 3 2 ( 4 n n ) for the partial sum. Now, using an easily found and rather well known upper bound for binomial coefficients, viz. ( n k ) < n k k ! , it is enough to show
3 2 ( 4 n ) n n ! < 2 4 n e n / 2
which is straightforward by induction.
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