User qifeng618

4 votes

How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?

answered Aug 30, 2021 at 10:02

3 votes

Direct combinatorial proof that $2^{2k} = \sum \binom{2i}{i}\binom{2j}{j}$?

answered Sep 10, 2021 at 13:11

2 votes

Accepted

Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?

answered Jan 23, 2023 at 15:53

1 vote

Accepted

Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?

answered Mar 25 at 9:31

1 vote

Binomial coefficient identity

answered Nov 20, 2021 at 6:03

1 vote

Ask for a reference or a proof of a combinatorial identity $\sum_{k=0}^n\binom{2n+1}{2k}\binom {k}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m}$

answered Nov 20, 2021 at 13:15

1 vote

Ask for a reference or a proof of an identity involving a finite sum and the Bernoulli numbers

answered Apr 17, 2022 at 12:29

1 vote

Ask for a proof of an identity involving the product of two Bernoulli numbers

answered Apr 17, 2022 at 12:57

1 vote

How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?

answered Sep 6, 2021 at 1:46

1 vote

Accepted

What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

answered Oct 6, 2021 at 18:00

0 votes

Accepted

How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?

answered Oct 9, 2021 at 11:27

0 votes

Does complete monotonicity of f imply log-concavity of f?

answered Sep 11, 2021 at 12:20

0 votes

Proof of complete monotonicity of a binomial function

answered Sep 11, 2021 at 14:47

0 votes

Polynomials akin to Bell polynomials

answered Sep 28, 2021 at 9:33

0 votes

Computing a sum involving factorials

answered Sep 28, 2021 at 14:12

0 votes

How to prove the convexity of a simple function involving a ratio of two polygamma functions?

answered Jul 6, 2021 at 2:15

0 votes

Accepted

What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?

answered Apr 10, 2023 at 2:41

0 votes

An identity for the ratio of two partial Bell polynomials

answered Oct 20, 2023 at 6:34

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18 Answers

How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?

Direct combinatorial proof that $2^{2k} = \sum \binom{2i}{i}\binom{2j}{j}$?

Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?

Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?

Binomial coefficient identity

Ask for a reference or a proof of a combinatorial identity $\sum_{k=0}^n\binom{2n+1}{2k}\binom {k}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m}$

Ask for a reference or a proof of an identity involving a finite sum and the Bernoulli numbers

Ask for a proof of an identity involving the product of two Bernoulli numbers

How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?

What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?

Does complete monotonicity of f imply log-concavity of f?

Proof of complete monotonicity of a binomial function

Polynomials akin to Bell polynomials

Computing a sum involving factorials

How to prove the convexity of a simple function involving a ratio of two polygamma functions?

What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?

An identity for the ratio of two partial Bell polynomials