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Proving $\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$

score 1k · Answer 1 · 2026-03-27 13:40:57

1k

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Proving $\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$

Published on 27 Mar 2026 - 13:40

score 46 · Answer 2 · 2026-03-30 01:45:08

46

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31

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score 1.5k · Answer 4 · 2019-03-30 14:39:03

1.5k

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score 34 · Answer 5 · 2019-03-30 18:13:52

34

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score 192 · Answer 6 · 2019-03-31 09:10:54

192

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score 58 · Answer 7 · 2026-03-30 20:53:48

58

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score 564 · Answer 8 · 2019-04-01 14:43:21

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51

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26

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score 96 · Answer 11 · 2019-04-02 03:02:37

96

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How to find $a$ & $b$ that satisfy $10^{n+1}=\sum_{i=1}^{n}{(a_{i}10^{i}-b_{i}9^{i})}$?

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score 104 · Answer 12 · 2026-03-25 06:17:35

104

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score 846 · Answer 13 · 2026-04-02 21:39:18

846

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score 99 · Answer 14 · 2019-04-02 09:59:06

99

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score 278 · Answer 15 · 2019-04-02 11:44:56

278

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List Question

Proving $\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$

What are the bounds for $e$?

Evaluating a (probably) arithmetic progression

Evaluate $\int_a^b\frac{1}{ x^2}dx$ using limit of a sum definition

An equality: from one sum to 2 sums.

Calculate $\sum_{n=1}^\infty \frac{n+1}{n(n+2)^2}$ using Basel Problem sum

In expressing arctangent as a series, why does substituting $x=1$ make sense?

sum of elements in a matrix

Finding the n-th partial sum of a series

Analytical expression for a sum into two loops

How to find $a$ & $b$ that satisfy $10^{n+1}=\sum_{i=1}^{n}{(a_{i}10^{i}-b_{i}9^{i})}$?

Evaluating a sum involving multinomial coefficient

How do you notate a sum of only some of the elements of an array?

Summation equality with finite differences - prove with induction

Dirac and sign errors in odd number of places

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