Question

Prove the identity $\sum\limits_{k=1}^{n+1} \sum\limits_{m=0}^{k-1} \binom{n+1}{k} \binom{n}{m} = 2^{2n}$

score 71 · Answer 1 · 2019-09-09 20:55:27

71

Views

Prove the identity $\sum\limits_{k=1}^{n+1} \sum\limits_{m=0}^{k-1} \binom{n+1}{k} \binom{n}{m} = 2^{2n}$

Published on 09 Sep 2019 - 20:55

score 56 · Answer 2 · 2019-09-09 22:52:03

56

Views

Simplification of the sum $\sum_{k=0}^M x^k\binom{M}{k}\binom{r}{k}?$

Published on 09 Sep 2019 - 22:52

score 72 · Answer 3 · 2019-09-10 17:54:02

72

Views

About the probability function $ \alpha \binom{x+q}{q}$

Published on 10 Sep 2019 - 17:54

score 182 · Answer 4 · 2026-03-29 18:33:25

182

Views

Is there a general formula to prove the equivalency of these two expansions of $~\dfrac{1}{(1-x)^2}~$?

Published on 29 Mar 2026 - 18:33

score 198 · Answer 5 · 2026-03-26 06:34:45

198

Views

Counting argument for $\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}\binom{n}{j}\frac{1}{k} = \frac{1}{n}$

Published on 26 Mar 2026 - 6:34

score 156 · Answer 6 · 2026-03-28 03:03:10

156

Views

Expression of $x^n+\frac1{x^n}$ by $x+\frac1{x}$ where $n$ is a positive odd number.

Published on 28 Mar 2026 - 3:03

score 185 · Answer 7 · 2019-09-12 21:42:58

185

Views

Compute $\sum \binom{n}{k} \binom{n+k-1}{k} (-1)^k$

Published on 12 Sep 2019 - 21:42

score 210 · Answer 8 · 2019-09-13 13:11:55

210

Views

Evaluate ${k \choose 0} + {k \choose 2} + {k \choose 4}+ {k \choose 6} +...+ {k \choose k}$ for even $k$

Published on 13 Sep 2019 - 13:11

score 39 · Answer 9 · 2019-09-14 07:03:03

39

Views

Coefficient of an expansion

Published on 14 Sep 2019 - 7:03

score 85 · Answer 10 · 2019-09-15 18:46:18

85

Views

Prove $n \choose m$ is divisible by n when $\gcd(m,n)=1$

Published on 15 Sep 2019 - 18:46

score 83 · Answer 11 · 2019-09-17 16:50:44

83

Views

Problem in Generalized Binomial coefficient

Published on 17 Sep 2019 - 16:50

score 35 · Answer 12 · 2019-09-17 17:10:12

35

Views

can we have $(\partial _x+f(x))^n=\partial _x^n+\binom{n}{1} (\partial _x)^{n-1}f(x)+\cdots+(f(x))^n$?

Published on 17 Sep 2019 - 17:10

score 85 · Answer 13 · 2019-09-17 21:59:53

85

Views

Asymptotic of sum with binomial coefficients.

Published on 17 Sep 2019 - 21:59

score 43 · Answer 14 · 2019-09-18 23:03:09

43

Views

Drawing a number of balls from $n$ numbered balls $k+1$ times with replacement

Published on 18 Sep 2019 - 23:03

score 38 · Answer 15 · 2019-09-19 07:22:41

38

Views

"If all the binomial coefficients (except the 1's) of $n$ are divisible by $n$, then $n$ is a prime no ".

Published on 19 Sep 2019 - 7:22

List Question

Prove the identity $\sum\limits_{k=1}^{n+1} \sum\limits_{m=0}^{k-1} \binom{n+1}{k} \binom{n}{m} = 2^{2n}$

Simplification of the sum $\sum_{k=0}^M x^k\binom{M}{k}\binom{r}{k}?$

About the probability function $ \alpha \binom{x+q}{q}$

Is there a general formula to prove the equivalency of these two expansions of $~\dfrac{1}{(1-x)^2}~$?

Counting argument for $\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}\binom{n}{j}\frac{1}{k} = \frac{1}{n}$

Expression of $x^n+\frac1{x^n}$ by $x+\frac1{x}$ where $n$ is a positive odd number.

Compute $\sum \binom{n}{k} \binom{n+k-1}{k} (-1)^k$

Evaluate ${k \choose 0} + {k \choose 2} + {k \choose 4}+ {k \choose 6} +...+ {k \choose k}$ for even $k$

Coefficient of an expansion

Prove $n \choose m$ is divisible by n when $\gcd(m,n)=1$

Problem in Generalized Binomial coefficient

can we have $(\partial _x+f(x))^n=\partial _x^n+\binom{n}{1} (\partial _x)^{n-1}f(x)+\cdots+(f(x))^n$?

Asymptotic of sum with binomial coefficients.

Drawing a number of balls from $n$ numbered balls $k+1$ times with replacement

"If all the binomial coefficients (except the 1's) of $n$ are divisible by $n$, then $n$ is a prime no ".

Trending Questions

Popular # Hahtags

Popular Questions