Question

Are these two binomial sums known? Proven generalization to the Hockey Stick patterns in Pascal's Triangle

score 1.7k · Answer 1 · 2014-03-15 20:28:15

1.7k

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Are these two binomial sums known? Proven generalization to the Hockey Stick patterns in Pascal's Triangle

Published on 15 Mar 2014 - 20:28

score 306 · Answer 2 · 2026-03-29 21:53:51

306

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Conditions for which $n | {n \choose k}$ for all $k$

Published on 29 Mar 2026 - 21:53

score 929 · Answer 3 · 2014-03-16 12:31:23

929

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Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$

Published on 16 Mar 2014 - 12:31

score 129 · Answer 4 · 2014-03-16 20:19:00

129

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Combinatoric identity $2n\times3^{n-1}+4n(n-1)3^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}2^k$

Published on 16 Mar 2014 - 20:19

score 1.4k · Answer 5 · 2026-03-26 09:26:58

1.4k

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Partial fraction expansion of $\frac{1}{x(x+1)(x+2)\cdots(x+n)}$

Published on 26 Mar 2026 - 9:26

score 1.6k · Answer 6 · 2014-03-17 23:03:12

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Remarkable and odd property, why is $\sum_{k=1}^i(-1)^{(i+k)}\text{ }^iC_kk^j= 0$ for $i>j$

Published on 17 Mar 2014 - 23:03

score 1.3k · Answer 7 · 2014-03-18 20:28:03

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Identity involving binomial coefficients: $\sum_{j=0}^i (-1)^{i+j}\binom {n-j}{i-j}\binom mj=\sum_{j=0}^i (-1)^{i+j}\binom {n-j+k}{i-j}\binom {m+k}j$

Published on 18 Mar 2014 - 20:28

score 760 · Answer 8 · 2014-03-18 22:39:34

760

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Prove $\binom{n}{m} \binom{m}{k} = \binom{n}{k} \binom{n-k}{m-k}$ by counting in two ways

Published on 18 Mar 2014 - 22:39

score 4.7k · Answer 9 · 2026-03-29 14:57:47

4.7k

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find the coefficient of the given term when the expression is expanded by the binomial theorem

Published on 29 Mar 2026 - 14:57

score 149 · Answer 10 · 2014-03-19 13:59:33

149

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How find this sum $\sum_{k=0}^{\left[\frac{n}{2}\right]}\frac{(-1)^k\binom{n-k}{k}}{n-k}$

Published on 19 Mar 2014 - 13:59

score 419 · Answer 11 · 2014-03-19 17:52:38

419

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Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$

Published on 19 Mar 2014 - 17:52

score 233 · Answer 12 · 2014-03-19 18:04:41

233

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binomial identity involving sum of diagonal elements

Published on 19 Mar 2014 - 18:04

score 115 · Answer 13 · 2026-03-28 05:00:46

115

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A combinatorics question on a sequence of binomial coefficents

Published on 28 Mar 2026 - 5:00

score 411 · Answer 14 · 2026-03-28 09:53:32

411

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Proving that $\sum_{k=0}^n\frac{1}{n\choose k}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}{k}$

Published on 28 Mar 2026 - 9:53

score 3.4k · Answer 15 · 2014-03-20 06:14:21

3.4k

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How to count matrices with rows and columns with an odd number of ones?

Published on 20 Mar 2014 - 6:14

List Question

Are these two binomial sums known? Proven generalization to the Hockey Stick patterns in Pascal's Triangle

Conditions for which $n | {n \choose k}$ for all $k$

Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$

Combinatoric identity $2n\times3^{n-1}+4n(n-1)3^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}2^k$

Partial fraction expansion of $\frac{1}{x(x+1)(x+2)\cdots(x+n)}$

Remarkable and odd property, why is $\sum_{k=1}^i(-1)^{(i+k)}\text{ }^iC_kk^j= 0$ for $i>j$

Identity involving binomial coefficients: $\sum_{j=0}^i (-1)^{i+j}\binom {n-j}{i-j}\binom mj=\sum_{j=0}^i (-1)^{i+j}\binom {n-j+k}{i-j}\binom {m+k}j$

Prove $\binom{n}{m} \binom{m}{k} = \binom{n}{k} \binom{n-k}{m-k}$ by counting in two ways

find the coefficient of the given term when the expression is expanded by the binomial theorem

How find this sum $\sum_{k=0}^{\left[\frac{n}{2}\right]}\frac{(-1)^k\binom{n-k}{k}}{n-k}$

Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$

binomial identity involving sum of diagonal elements

A combinatorics question on a sequence of binomial coefficents

Proving that $\sum_{k=0}^n\frac{1}{n\choose k}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}{k}$

How to count matrices with rows and columns with an odd number of ones?

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