Question

Alternate proof of $\sum_{j=0}^{m}\sum_{k=0}^{n}\left(-1\right)^{j+k}\binom{j+k}{k}\binom{m}{j}\binom{n}{k}=\delta_{mn}$

score 172 · Answer 1 · 2026-03-30 10:41:02

172

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Alternate proof of $\sum_{j=0}^{m}\sum_{k=0}^{n}\left(-1\right)^{j+k}\binom{j+k}{k}\binom{m}{j}\binom{n}{k}=\delta_{mn}$

Published on 30 Mar 2026 - 10:41

score 66 · Answer 2 · 2019-11-20 05:57:26

66

Views

Value of expression $\sum\limits^{10}_{r=2}\binom{r}{2}\cdot \binom{10}{r}$

Published on 20 Nov 2019 - 5:57

score 43 · Answer 3 · 2019-11-20 10:26:32

43

Views

Sum over subsets of a function of binomial coefficients

Published on 20 Nov 2019 - 10:26

score 126 · Answer 4 · 2026-03-26 12:57:58

126

Views

Combinatorial proof for identity $\sum_{k=0}^{2n}(-1)^k\binom{3n}{k}=\binom{3n-1}{2n}$

Published on 26 Mar 2026 - 12:57

score 69 · Answer 5 · 2019-11-21 15:48:24

69

Views

Non-trivial alternating sums of binomial coefficients

Published on 21 Nov 2019 - 15:48

score 54 · Answer 6 · 2026-03-26 12:53:30

54

Views

Is there a combinatorial proof of the Taylor series $\frac1{\sqrt{1-4x}}=\sum_{n=0}^\infty{2n\choose n}x^n$?

Published on 26 Mar 2026 - 12:53

score 132 · Answer 7 · 2019-11-22 01:32:15

132

Views

Number theory and binomial coefficients

Published on 22 Nov 2019 - 1:32

score 220 · Answer 8 · 2026-04-02 17:36:07

220

Views

Prove that $x^2+px+p^2$ is a factor $(x+p)^n-x^n-p^n$, if $n$ be odd and not divisible by $3$.

Published on 02 Apr 2026 - 17:36

score 41 · Answer 9 · 2019-11-24 18:58:36

41

Views

Prove or give reference for $\int \frac{1}{\sum _{k=1}^n x^{k-1} \binom{n-1}{k-1}} \, dx=\frac{(x+1)^{2-n}}{2-n}$

Published on 24 Nov 2019 - 18:58

score 120 · Answer 10 · 2019-11-26 18:20:37

120

Views

How many ways to group $5$ into two pairs and a single? I get $30$, but the answer is $15$. How can I tell that I over-counted?

Published on 26 Nov 2019 - 18:20

score 327 · Answer 11 · 2019-11-27 09:21:38

327

Views

Find the coefficient of $\frac{1}{n^4}$ in the given Summation.

Published on 27 Nov 2019 - 9:21

score 68 · Answer 12 · 2026-04-06 13:48:38

68

Views

correct property of a Gamma function?

Published on 06 Apr 2026 - 13:48

score 202 · Answer 13 · 2026-03-29 13:27:48

202

Views

Interpretation of a version of the Binomial Coefficient with double factorial

Published on 29 Mar 2026 - 13:27

score 249 · Answer 14 · 2026-02-23 10:22:04

249

Views

How to actually calculate the probability to have extracted each one of $n$ balls at least once after $k$ turns?

Published on 23 Feb 2026 - 10:22

score 64 · Answer 15 · 2019-11-28 21:40:53

64

Views

Show that $\sum_{k=0}^n \binom{m-k}{n-k} = \binom{m+1}{n}$

Published on 28 Nov 2019 - 21:40

List Question

Alternate proof of $\sum_{j=0}^{m}\sum_{k=0}^{n}\left(-1\right)^{j+k}\binom{j+k}{k}\binom{m}{j}\binom{n}{k}=\delta_{mn}$

Value of expression $\sum\limits^{10}_{r=2}\binom{r}{2}\cdot \binom{10}{r}$

Sum over subsets of a function of binomial coefficients

Combinatorial proof for identity $\sum_{k=0}^{2n}(-1)^k\binom{3n}{k}=\binom{3n-1}{2n}$

Non-trivial alternating sums of binomial coefficients

Is there a combinatorial proof of the Taylor series $\frac1{\sqrt{1-4x}}=\sum_{n=0}^\infty{2n\choose n}x^n$?

Number theory and binomial coefficients

Prove that $x^2+px+p^2$ is a factor $(x+p)^n-x^n-p^n$, if $n$ be odd and not divisible by $3$.

Prove or give reference for $\int \frac{1}{\sum _{k=1}^n x^{k-1} \binom{n-1}{k-1}} \, dx=\frac{(x+1)^{2-n}}{2-n}$

How many ways to group $5$ into two pairs and a single? I get $30$, but the answer is $15$. How can I tell that I over-counted?

Find the coefficient of $\frac{1}{n^4}$ in the given Summation.

correct property of a Gamma function?

Interpretation of a version of the Binomial Coefficient with double factorial

How to actually calculate the probability to have extracted each one of $n$ balls at least once after $k$ turns?

Show that $\sum_{k=0}^n \binom{m-k}{n-k} = \binom{m+1}{n}$

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