Question

For $1\leq j\leq i\leq n$, does $\sum_{k=j}^i (-1)^{i-k} \binom{n-k}{i-k} \binom{n-j}{k-j}$ equal $\delta_{i,j}$?

score 109 · Answer 1 · 2016-09-22 16:25:10

109

Views

For $1\leq j\leq i\leq n$, does $\sum_{k=j}^i (-1)^{i-k} \binom{n-k}{i-k} \binom{n-j}{k-j}$ equal $\delta_{i,j}$?

Published on 22 Sep 2016 - 16:25

score 274 · Answer 2 · 2026-04-08 17:35:54

274

Views

Finding an equation to a really complex sum

Published on 08 Apr 2026 - 17:35

score 268 · Answer 3 · 2016-09-23 03:06:04

268

Views

How do I re-index $\sum_{i=0}^t \sum_{j=0}^t \sum_{k=0}^\infty \binom{t}{i} \binom{t}{j} \binom{k+t-1}{t-1} x^{i+j+k}$

Published on 23 Sep 2016 - 3:06

score 657 · Answer 4 · 2016-09-23 14:50:02

657

Views

Combinatorial sum identity for a choose function $\sum\limits_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$

Published on 23 Sep 2016 - 14:50

score 1k · Answer 5 · 2026-03-27 15:14:10

1k

Views

$\sum_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$ using Counting argument

Published on 27 Mar 2026 - 15:14

score 4.2k · Answer 6 · 2016-09-24 14:14:29

4.2k

Views

Prove that $\sum_{k=0}^nk{m+k \choose m}=n{m+n+1\choose m+1}-{m+n+1 \choose m+2}$

Published on 24 Sep 2016 - 14:14

score 141 · Answer 7 · 2026-03-31 08:15:59

141

Views

In the expansion of $(1+x+x^2)^n$, find the required value

Published on 31 Mar 2026 - 8:15

score 104 · Answer 8 · 2026-03-31 10:42:53

104

Views

How to prove that $\sum\limits_{k=0}^{n} \binom{n+1}{k+1} = 2^{n+1} - 1$ using the Binomial Theorem?

Published on 31 Mar 2026 - 10:42

score 246 · Answer 9 · 2026-04-10 07:00:31

246

Views

Finding the maximum of a ratio of binomial coefficients

Published on 10 Apr 2026 - 7:00

score 175 · Answer 10 · 2016-09-25 21:05:20

175

Views

Proving by induction, involving a binomial coefficient and an inequality.

Published on 25 Sep 2016 - 21:05

score 205 · Answer 11 · 2026-03-27 15:09:52

205

Views

Summation of binomial with factor squared $\sum_{k=0}^n k^2 \binom nk^2 = n^2 {2n-2 \choose n-1}$

Published on 27 Mar 2026 - 15:09

score 508 · Answer 12 · 2016-09-26 01:37:58

508

Views

Series of binomial coefficients: $\sum\limits_{n=k}^{\infty}{\binom nk}x^n=\frac{x^k}{(1-x)^{k+1}}$

Published on 26 Sep 2016 - 1:37

score 1.9k · Answer 13 · 2016-09-26 10:01:39

1.9k

Views

Derivative of the binomial $\binom x n$ with respect to $x$

Published on 26 Sep 2016 - 10:01

score 2.5k · Answer 14 · 2016-09-26 11:58:42

2.5k

Views

Sum of Binomial Coefficients Times a Polynomial

Published on 26 Sep 2016 - 11:58

score 622 · Answer 15 · 2016-09-26 13:49:34

622

Views

Why is ${2n \choose n}/2 $ odd if and only if $n=2^k$?

Published on 26 Sep 2016 - 13:49

List Question

For $1\leq j\leq i\leq n$, does $\sum_{k=j}^i (-1)^{i-k} \binom{n-k}{i-k} \binom{n-j}{k-j}$ equal $\delta_{i,j}$?

Finding an equation to a really complex sum

How do I re-index $\sum_{i=0}^t \sum_{j=0}^t \sum_{k=0}^\infty \binom{t}{i} \binom{t}{j} \binom{k+t-1}{t-1} x^{i+j+k}$

Combinatorial sum identity for a choose function $\sum\limits_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$

$\sum_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$ using Counting argument

Prove that $\sum_{k=0}^nk{m+k \choose m}=n{m+n+1\choose m+1}-{m+n+1 \choose m+2}$

In the expansion of $(1+x+x^2)^n$, find the required value

How to prove that $\sum\limits_{k=0}^{n} \binom{n+1}{k+1} = 2^{n+1} - 1$ using the Binomial Theorem?

Finding the maximum of a ratio of binomial coefficients

Proving by induction, involving a binomial coefficient and an inequality.

Summation of binomial with factor squared $\sum_{k=0}^n k^2 \binom nk^2 = n^2 {2n-2 \choose n-1}$

Series of binomial coefficients: $\sum\limits_{n=k}^{\infty}{\binom nk}x^n=\frac{x^k}{(1-x)^{k+1}}$

Derivative of the binomial $\binom x n$ with respect to $x$

Sum of Binomial Coefficients Times a Polynomial

Why is ${2n \choose n}/2 $ odd if and only if $n=2^k$?

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