Question

How can it be shown that $\sum_{k=0}^{n}\binom{n}{k}\binom{2n-k}{n}\left(a-1\right)^k=\sum_{k=0}^{n}\binom{n}{k}^{2}a^{n-k}$

score 71 · Answer 1 · 2026-04-17 12:44:48

71

Views

How can it be shown that $\sum_{k=0}^{n}\binom{n}{k}\binom{2n-k}{n}\left(a-1\right)^k=\sum_{k=0}^{n}\binom{n}{k}^{2}a^{n-k}$

Published on 17 Apr 2026 - 12:44

score 104 · Answer 2 · 2026-04-12 03:36:08

104

Views

Closed form for $\sum_{k=0}^{n}\binom{n}{k}^{r}$

Published on 12 Apr 2026 - 3:36

score 108 · Answer 3 · 2026-04-12 19:13:57

108

Views

Evaluating the sum $\sum\limits_{n=1}^{\infty}\frac{x^{2n}}{(2n+1)!}$

Published on 12 Apr 2026 - 19:13

score 266 · Answer 4 · 2026-03-25 11:09:02

266

Views

Surprising fact about a certain number-theoretic function

Published on 25 Mar 2026 - 11:09

score 52 · Answer 5 · 2026-04-10 14:10:51

52

Views

double summation problem, index change formula

Published on 10 Apr 2026 - 14:10

score 365 · Answer 6 · 2026-04-12 05:24:35

365

Views

An example to show that convergence of Cesaro sum to $0$ does not imply the original sequence converges to $0$.

Published on 12 Apr 2026 - 5:24

score 247 · Answer 7 · 2026-04-17 02:48:38

247

Views

Evaluate the sum $\binom{2n}{n}+\binom{3n}{n}+\binom{4n}{n}+\cdots+\binom{kn}{n}$

Published on 17 Apr 2026 - 2:48

score 107 · Answer 8 · 2026-04-16 19:57:58

107

Views

Prove $\sum_{k=0}^{r}\binom{n}{k}\binom{n}{r-k}\left(-1\right)^{k}=\left(-1\right)^{\frac{r}{2}}\binom{n}{\frac{r}{2}}$ for $r$ even.

Published on 16 Apr 2026 - 19:57

score 64 · Answer 9 · 2026-04-09 07:54:09

64

Views

Rearrange a Triple Summation with constraints

Published on 09 Apr 2026 - 7:54

score 104 · Answer 10 · 2026-04-12 18:03:31

104

Views

Leading order behaviour of an infinite sum

Published on 12 Apr 2026 - 18:03

score 297 · Answer 11 · 2026-04-15 05:16:13

297

Views

Solving binomial summation $\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n-k}{k} 2^{n-k}$

Published on 15 Apr 2026 - 5:16

score 180 · Answer 12 · 2026-04-13 20:29:46

180

Views

Looking for a short proof of a harmless looking binomial identity

Published on 13 Apr 2026 - 20:29

score 45 · Answer 13 · 2026-04-13 02:50:11

45

Views

How do I find all the different cases of convergence/divergence of this series?

Published on 13 Apr 2026 - 2:50

score 87 · Answer 14 · 2026-04-13 07:51:54

87

Views

Summation problem (associated with use of Stirling's Approx)

Published on 13 Apr 2026 - 7:51

score 214 · Answer 15 · 2026-04-29 02:17:17

214

Views

Interesting patterns related to the sums of the remainders of integers

Published on 29 Apr 2026 - 2:17

List Question

How can it be shown that $\sum_{k=0}^{n}\binom{n}{k}\binom{2n-k}{n}\left(a-1\right)^k=\sum_{k=0}^{n}\binom{n}{k}^{2}a^{n-k}$

Closed form for $\sum_{k=0}^{n}\binom{n}{k}^{r}$

Evaluating the sum $\sum\limits_{n=1}^{\infty}\frac{x^{2n}}{(2n+1)!}$

Surprising fact about a certain number-theoretic function

double summation problem, index change formula

An example to show that convergence of Cesaro sum to $0$ does not imply the original sequence converges to $0$.

Evaluate the sum $\binom{2n}{n}+\binom{3n}{n}+\binom{4n}{n}+\cdots+\binom{kn}{n}$

Prove $\sum_{k=0}^{r}\binom{n}{k}\binom{n}{r-k}\left(-1\right)^{k}=\left(-1\right)^{\frac{r}{2}}\binom{n}{\frac{r}{2}}$ for $r$ even.

Rearrange a Triple Summation with constraints

Leading order behaviour of an infinite sum

Solving binomial summation $\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n-k}{k} 2^{n-k}$

Looking for a short proof of a harmless looking binomial identity

How do I find all the different cases of convergence/divergence of this series?

Summation problem (associated with use of Stirling's Approx)

Interesting patterns related to the sums of the remainders of integers

Trending Questions

Popular # Hahtags

Popular Questions