List questions from binomial-coefficients

Published on 18 Dec 2018 - 3:07

169

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Is there a closed formula for $\sum_{k=0}^n \binom{\alpha}{n-k} \binom{\beta}{k}(-1)^k$?

Published on 18 Dec 2018 - 23:55

93

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Prove $\sum_{k=0}^n \binom{m+k}{m} = \binom{m+n+1}{m+1}$ using the binomium of Newton

Published on 19 Dec 2018 - 8:06

56

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Binomial sum for an arbitrary function

Published on 24 Mar 2026 - 22:09

214

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Prove that $\sum_{k=0}^{2n} \binom {2n+k}{k} \binom{2n}{k} \frac{(-1)^k}{2^k} \frac{1}{k+1} = 0. $

Published on 19 Dec 2018 - 19:05

139

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Find the determinant of the $n\times n$ matrix $A_n$ with $(A_n)_{i,j}={n\choose |i-j|}$.

Published on 19 Dec 2018 - 19:12

240

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Prove $\sum_{k=m}^n{k\choose k-m}{2n\choose 2k}=4^{n-m}\frac{n(2n-m-1)!}{(2n-2m)!m!}.$

Published on 20 Dec 2018 - 14:00

77

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What are the prime divisors of $\det(A_n)$, where $A_n$ is the $n\times n$ matrix given by $(A_n)_{i,j}={n\choose|i-j|}$?

Published on 25 Mar 2026 - 20:12

188

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sum which have binomial coefficients

Published on 22 Dec 2018 - 10:21

289

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why is $\frac{k\cdot n!}{k!(n-k)!} = \frac{n(n-1)!}{(k-1)!((n-1)-(k-1))!}$?

Published on 26 Mar 2026 - 7:35

512

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Identity involving difference of binomial coefficients

Published on 25 Mar 2026 - 20:52

128

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simplify summation with binomial coefficients

Published on 22 Dec 2018 - 17:04

788

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How to simplify this binomial coefficients?

Published on 24 Dec 2018 - 9:57

122

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Double Binomial Sum

Published on 24 Dec 2018 - 17:10

541

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Prove that $\sum_{k = 0}^{49}(-1)^k\binom{99}{2k} = -2^{49}$

Published on 27 Mar 2026 - 3:41

List Question

Show $241|\binom{240}{120}$.

Is there a closed formula for $\sum_{k=0}^n \binom{\alpha}{n-k} \binom{\beta}{k}(-1)^k$?

Prove $\sum_{k=0}^n \binom{m+k}{m} = \binom{m+n+1}{m+1}$ using the binomium of Newton

Binomial sum for an arbitrary function

Prove that $\sum_{k=0}^{2n} \binom {2n+k}{k} \binom{2n}{k} \frac{(-1)^k}{2^k} \frac{1}{k+1} = 0. $

Find the determinant of the $n\times n$ matrix $A_n$ with $(A_n)_{i,j}={n\choose |i-j|}$.

Prove $\sum_{k=m}^n{k\choose k-m}{2n\choose 2k}=4^{n-m}\frac{n(2n-m-1)!}{(2n-2m)!m!}.$

What are the prime divisors of $\det(A_n)$, where $A_n$ is the $n\times n$ matrix given by $(A_n)_{i,j}={n\choose|i-j|}$?

sum which have binomial coefficients

why is $\frac{k\cdot n!}{k!(n-k)!} = \frac{n(n-1)!}{(k-1)!((n-1)-(k-1))!}$?

Identity involving difference of binomial coefficients

simplify summation with binomial coefficients

How to simplify this binomial coefficients?

Double Binomial Sum

Prove that $\sum_{k = 0}^{49}(-1)^k\binom{99}{2k} = -2^{49}$

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