Question

I have this identity that I'd like to prove. $\sum_{k=0}^{n}\left(\frac{n-2k}{n}\binom{n}{k}\right)^2=\frac{2}{n}\binom{2n-2}{n-1}$

score 110 · Answer 1 · 2026-04-01 22:19:03

110

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I have this identity that I'd like to prove. $\sum_{k=0}^{n}\left(\frac{n-2k}{n}\binom{n}{k}\right)^2=\frac{2}{n}\binom{2n-2}{n-1}$

Published on 01 Apr 2026 - 22:19

score 96 · Answer 2 · 2026-03-25 18:53:27

96

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Converting recurrence involving binomial terms and generating functions to closed form.

Published on 25 Mar 2026 - 18:53

score 39 · Answer 3 · 2026-04-01 22:18:05

39

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Why does $\sum_{i=n}^{2n-1}\binom{i-1}{n-1}2^{1-i}$ computes the probability of $n$ head or tails

Published on 01 Apr 2026 - 22:18

score 157 · Answer 4 · 2020-06-18 19:30:45

157

Views

Show that ${n\choose k} = {n\choose l} \iff n=k+l$

Published on 18 Jun 2020 - 19:30

score 162 · Answer 5 · 2020-06-19 18:27:45

162

Views

Find the value of $a_{0}^{2}-a_{1}^{2}+a_{2}^{2}- \dots+a_{2 n}^{2}$

Published on 19 Jun 2020 - 18:27

score 239 · Answer 6 · 2020-06-20 05:06:17

239

Views

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Published on 20 Jun 2020 - 5:06

score 159 · Answer 7 · 2026-03-30 14:43:32

159

Views

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Published on 30 Mar 2026 - 14:43

score 480 · Answer 8 · 2020-06-20 20:30:05

480

Views

Remainder when a binomial coefficient is divided by a prime

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score 135 · Answer 9 · 2026-03-30 14:39:05

135

Views

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score 254 · Answer 10 · 2026-02-23 10:16:53

254

Views

Double sum identity involving binomial coefficients, possibly connected to umbral calculus

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score 118 · Answer 11 · 2026-03-26 09:51:25

118

Views

Showing $\sum_{k=0}^{n+1} \binom n k \frac{(-1)^k}{(n+k)(n+k+1)} = \sum_{k=0}^{n+1} \binom {n+1} k \frac{ (-1)^k}{n+k}$

Published on 26 Mar 2026 - 9:51

score 36 · Answer 12 · 2020-06-22 08:39:37

36

Views

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score 33 · Answer 13 · 2026-03-28 10:54:19

33

Views

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score 124 · Answer 14 · 2026-03-30 14:37:44

124

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Published on 30 Mar 2026 - 14:37

score 67 · Answer 15 · 2020-06-24 07:56:45

67

Views

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List Question

I have this identity that I'd like to prove. $\sum_{k=0}^{n}\left(\frac{n-2k}{n}\binom{n}{k}\right)^2=\frac{2}{n}\binom{2n-2}{n-1}$

Converting recurrence involving binomial terms and generating functions to closed form.

Why does $\sum_{i=n}^{2n-1}\binom{i-1}{n-1}2^{1-i}$ computes the probability of $n$ head or tails

Show that ${n\choose k} = {n\choose l} \iff n=k+l$

Find the value of $a_{0}^{2}-a_{1}^{2}+a_{2}^{2}- \dots+a_{2 n}^{2}$

Values for partial row starts of Pascal's triangle

How is $nq-1 = \sum_{m=0}^n (m-1){n \choose m} p^{n-m}q^m$

Remainder when a binomial coefficient is divided by a prime

Source for similarity between Leibniz formula and binomial theorem?

Double sum identity involving binomial coefficients, possibly connected to umbral calculus

Showing $\sum_{k=0}^{n+1} \binom n k \frac{(-1)^k}{(n+k)(n+k+1)} = \sum_{k=0}^{n+1} \binom {n+1} k \frac{ (-1)^k}{n+k}$

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