Question

Show that $\sum_{k=0}^{n+1}\left(\binom{n}{k}-\binom{n}{k-1}\right)^2 = \frac{2}{n+1}\binom{2n}{n}$

score 115 · Answer 1 · 2026-03-29 11:50:12

115

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Show that $\sum_{k=0}^{n+1}\left(\binom{n}{k}-\binom{n}{k-1}\right)^2 = \frac{2}{n+1}\binom{2n}{n}$

Published on 29 Mar 2026 - 11:50

score 61 · Answer 2 · 2019-10-20 10:04:36

61

Views

Proofing $\sum^n_{k=0}(-1)^k{n\choose k}(a_1+kd)=0$

Published on 20 Oct 2019 - 10:04

score 139 · Answer 3 · 2026-03-28 05:21:53

139

Views

Basic Number Theory Inequality assuming Bertrand's postulate false for some n.

Published on 28 Mar 2026 - 5:21

score 150 · Answer 4 · 2026-02-23 06:59:34

150

Views

What is the max length string that can be formed using k distinct characters so that all of its substrings are unique.

Published on 23 Feb 2026 - 6:59

score 577 · Answer 5 · 2026-03-27 10:07:04

577

Views

Rewriting a binomial formula summation

Published on 27 Mar 2026 - 10:07

score 73 · Answer 6 · 2019-10-23 19:13:08

73

Views

Cannot find the mistake in a probability question

Published on 23 Oct 2019 - 19:13

score 160 · Answer 7 · 2026-03-27 04:58:56

160

Views

Counting the ways on grid if one can move from $(x,y)$ to $(x+a, x+b)$ for arbitrary $x,y,a,b\geq 0$.

Published on 27 Mar 2026 - 4:58

score 453 · Answer 8 · 2026-03-30 06:46:39

453

Views

Finding a closed form for (a) $\sum_{k=0}^n 4^k {2n \choose 2k}$

Published on 30 Mar 2026 - 6:46

score 87 · Answer 9 · 2019-10-24 10:56:47

87

Views

Splittings for a distinct pack of cards (without $\heartsuit,\spadesuit,\diamondsuit,\clubsuit $)

Published on 24 Oct 2019 - 10:56

score 204 · Answer 10 · 2019-10-24 14:34:10

204

Views

Coefficients of (1+x)(1+2x)...(1+nx)

Published on 24 Oct 2019 - 14:34

score 261 · Answer 11 · 2026-03-26 13:01:22

261

Views

Combinatorial Proof of $\binom{n+1}{k} = (n+1)\binom{n}{k-1}/k$

Published on 26 Mar 2026 - 13:01

score 83 · Answer 12 · 2019-10-25 05:24:04

Prove $\sum_{i=0}^{\min(x,y)}\binom{x}{i}\binom{y}{i}2^{x+y-(i+1)}=\sum_{i=0}^{\min(x,y)}2^{x+y-(i+1)} (-1)^i \frac{(x+y-i)!}{(x-i)!(y-i)!i!}$

score 76 · Answer 13 · 2019-10-25 17:25:39

76

Views

Digits patterns in power number

Published on 25 Oct 2019 - 17:25

score 534 · Answer 14 · 2019-10-26 06:44:47

534

Views

How to find the coefficient of $x^3y^5$ in the expression $(1+xy+y^2)^n$

Published on 26 Oct 2019 - 6:44

score 63 · Answer 15 · 2019-10-26 13:09:27

63

Views

Use the definition to prove that $\sum_{j=0}^n \binom{n}{j}=2^n $

Published on 26 Oct 2019 - 13:09

List Question

Show that $\sum_{k=0}^{n+1}\left(\binom{n}{k}-\binom{n}{k-1}\right)^2 = \frac{2}{n+1}\binom{2n}{n}$

Proofing $\sum^n_{k=0}(-1)^k{n\choose k}(a_1+kd)=0$

Basic Number Theory Inequality assuming Bertrand's postulate false for some n.

What is the max length string that can be formed using k distinct characters so that all of its substrings are unique.

Rewriting a binomial formula summation

Cannot find the mistake in a probability question

Counting the ways on grid if one can move from $(x,y)$ to $(x+a, x+b)$ for arbitrary $x,y,a,b\geq 0$.

Finding a closed form for (a) $\sum_{k=0}^n 4^k {2n \choose 2k}$

Splittings for a distinct pack of cards (without $\heartsuit,\spadesuit,\diamondsuit,\clubsuit $)

Coefficients of (1+x)(1+2x)...(1+nx)

Combinatorial Proof of $\binom{n+1}{k} = (n+1)\binom{n}{k-1}/k$

Prove $\sum_{i=0}^{\min(x,y)}\binom{x}{i}\binom{y}{i}2^{x+y-(i+1)}=\sum_{i=0}^{\min(x,y)}2^{x+y-(i+1)} (-1)^i \frac{(x+y-i)!}{(x-i)!(y-i)!i!}$

Digits patterns in power number

How to find the coefficient of $x^3y^5$ in the expression $(1+xy+y^2)^n$

Use the definition to prove that $\sum_{j=0}^n \binom{n}{j}=2^n $

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