4.247 binomial-coefficients questions.

On the integral $-\int\log\left(\binom{1-x}{6}+\binom{x}{6}\right)dx$, and its definite integral over the unit interval

While I was playing with Wolfram Alpha online calculator I wondered that I know how to calculate with the help of this tool and my knowledges the first cases for integers $n\geq 1$ of this type of ...

-2 Solving in terms of binomial coefficient [on hold]

1 answers, 46 views combinatorics binomial-coefficients
I am figuring out a possible pattern which somehow follows the pascal triangle. Suppose we have the following: $${{a}\choose{b}}+{{c}\choose{d}}+{{e}\choose{f}}+{{g}\choose{h}}={{8}\choose{1}}$$ I ...

3 Periodic sequences resulting from a summation over the Thue–Morse sequence

Let $s_2(n)$ denote the sum of digits of $n$ in base-2 (OEIS sequence A000120), and $t_n=(-1)^{s_2(n)}$. Note that $t_n$ is the signed Thue–Morse sequence (OEIS sequence A106400), satisfying the ...

Could someone please explain the steps to solve an equation like this?

Honestly, I have no idea if I put the correct tag on this question, and I don't even know where to begin to solve an equation like this: $$f(d,n)=\sum_{i=1}^n\binom{d}{i}.$$ Could someone explain ...

-1 Binomial Expansions No calculator

‘Show how the binomial expansion can be used to work out $268^2 - 232^2$ without a calculator.’ Also to work out 469 * 548 + 469 * 17 without a calculator. I understand the process of binomial ...

8 Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $[q:1,1,1,1..1,2,2,..2]$. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...

1 sum of alternating binomial

Compute the sum $\sum_{k=0}^{n}(-1)^k k^n\binom{n}{k}$ I've seen a solution along the following lines here, page 3: Consider $(1+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k$. ($\star$) We prove by ...

-1 Relation between sum of combinations

2 answers, 38 views combinatorics binomial-coefficients
I just realised the following identity: $$\sum_{i=0}^n {2n+1\choose i}=\sum_{i=0}^{2n}{2n\choose i}.$$ Is that correct? Has this been already proved?

-1 Evaluate the Binomial Sum [closed]

Evaluate the following sum involving binomial coefficients, $$\sum_{k=0}^n \binom{n}{k}\frac{k!}{(n+1+k)!}$$ when $n$ is a nonnegative integer.

1 The series$\sum\limits_{n=1}^{\infty}(\frac{(2n-1)!!}{(2n)!!})^p$ converges if and only if $p>2$

Problem: The series$\sum\limits_{n=1}^{\infty}(\frac{(2n-1)!!}{(2n)!!})^p$ converges if and only if $p>2$ My question is: can anyone solve the problem by using binomial coefficients or Stirling's ...

Finding a similar formula for the more general product $x^{m}y^{n}$

show that if $xy=ax+by$ then $$x^{n}y^{n}=\sum_{k=1}^{n}\binom{2n-1-k}{n-1}(a^{n}b^{n-k}x^{k}+a^{n-k}b^{n}y^{k})\quad\quad for\:all\quad n>0.$$ Finding a similar formula for the more general ...

Finding a closed form for this binomial coefficients summation term

I am trying to find the closed form for this binomial summation term. $$\sum_{j=1}^{m}(-1)^{j+1}\binom{r}{j}\sum_{k=1}^{n}\binom{-j+rk+s}{m-j}, \quad\quad\quad integers\quad m,n\geq0.$$ Need ...

1 Asymptotic expansion of a sum containing binomial coefficients

I'm trying to find an asymptotic expansion for the following sum: $$\sum_{k=2}^{n}\frac{n!}{k\left(n-k\right)!}=\sum_{k=2}^{n}\left(\begin{array}{c} n\\ k \end{array}\right)\left(k-1\right)!$$ for ...