Question

Prove that $\sum_{k=0}^{\lfloor (n-1)/2 \rfloor} (-1)^k {n+1 \choose k} {2n-2k-1 \choose n} =\frac{ n(n+1)}2 $

score 259 · Answer 1 · 2026-03-26 20:41:53

259

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Prove that $\sum_{k=0}^{\lfloor (n-1)/2 \rfloor} (-1)^k {n+1 \choose k} {2n-2k-1 \choose n} =\frac{ n(n+1)}2 $

Published on 26 Mar 2026 - 20:41

score 252 · Answer 2 · 2026-03-29 16:36:08

252

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Proof that the power of $2$ in $(3n)!$ is greater than or equals to the power of $2$ in $n!(n+1)!(n+2)!$

Published on 29 Mar 2026 - 16:36

score 157 · Answer 3 · 2020-01-05 07:44:11

157

Views

Prove that $\left( \sum_{j=0}^{\left[n/2\right]} {n \choose j} \left(1+p\right)^j \right)^{1/n}$ is increasing in $n$

Published on 05 Jan 2020 - 7:44

score 167 · Answer 4 · 2026-03-31 13:07:14

167

Views

Counting votes: How many possible paths exist on which candidate A is never 2 votes ahead

Published on 31 Mar 2026 - 13:07

score 126 · Answer 5 · 2020-01-07 03:35:06

126

Views

The sum of fractions of square of binomial coefficient DIVIDED by index variable

Published on 07 Jan 2020 - 3:35

score 170 · Answer 6 · 2020-01-07 09:58:23

170

Views

sum of coefficient of all even power of $x$

Published on 07 Jan 2020 - 9:58

score 145 · Answer 7 · 2020-01-07 23:32:34

145

Views

Calculate a sum of fraction of binomial coefficients

Published on 07 Jan 2020 - 23:32

score 97 · Answer 8 · 2026-03-29 18:17:24

97

Views

Is $\frac{(3n)!}{n!(n+1)!(n+2)!}$ integer for any integer $n \ge 3$?

Published on 29 Mar 2026 - 18:17

score 5.1k · Answer 9 · 2026-03-29 19:30:33

5.1k

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What is the coefficient of $x^4$ in the trinomial expansion of $(1+x+x^2)^{10}$?

Published on 29 Mar 2026 - 19:30

score 75 · Answer 10 · 2026-03-29 22:06:46

75

Views

Proving recursive function outputs $2^x \cdot {x \choose y}$

Published on 29 Mar 2026 - 22:06

score 173 · Answer 11 · 2020-01-10 16:31:14

173

Views

Closed form summation of expression with binomial coefficient

Published on 10 Jan 2020 - 16:31

score 76 · Answer 12 · 2020-01-11 10:53:35

76

Views

Prove $\sum\limits_{k=0}^{n}2^k \binom{n}{k}=3^n$

Published on 11 Jan 2020 - 10:53

score 270 · Answer 13 · 2020-01-12 12:05:59

270

Views

maximum of a product of binomial coefficients

Published on 12 Jan 2020 - 12:05

score 208 · Answer 14 · 2026-03-26 20:38:52

208

Views

How should I solve combination addition like this?

Published on 26 Mar 2026 - 20:38

score 50 · Answer 15 · 2020-01-12 18:23:18

50

Views

Binomial Coefficients Summation $\sum_{k=0}^n a^k \binom{n}{k}^2$

Published on 12 Jan 2020 - 18:23

List Question

Prove that $\sum_{k=0}^{\lfloor (n-1)/2 \rfloor} (-1)^k {n+1 \choose k} {2n-2k-1 \choose n} =\frac{ n(n+1)}2 $

Proof that the power of $2$ in $(3n)!$ is greater than or equals to the power of $2$ in $n!(n+1)!(n+2)!$

Prove that $\left( \sum_{j=0}^{\left[n/2\right]} {n \choose j} \left(1+p\right)^j \right)^{1/n}$ is increasing in $n$

Counting votes: How many possible paths exist on which candidate A is never 2 votes ahead

The sum of fractions of square of binomial coefficient DIVIDED by index variable

sum of coefficient of all even power of $x$

Calculate a sum of fraction of binomial coefficients

Is $\frac{(3n)!}{n!(n+1)!(n+2)!}$ integer for any integer $n \ge 3$?

What is the coefficient of $x^4$ in the trinomial expansion of $(1+x+x^2)^{10}$?

Proving recursive function outputs $2^x \cdot {x \choose y}$

Closed form summation of expression with binomial coefficient

Prove $\sum\limits_{k=0}^{n}2^k \binom{n}{k}=3^n$

maximum of a product of binomial coefficients

How should I solve combination addition like this?

Binomial Coefficients Summation $\sum_{k=0}^n a^k \binom{n}{k}^2$

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