Newbetuts

Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-

Coefficient extraction

Evaluating $\sum_{k=0}^n \binom{n}{k} 2^{k^2}$

Derive a closed form for a sum with inverse binomial coefficients

Intuition behind negative combinations

How do I prove this combinatorial identity using inclusion and exclusion principle?

Prove that $\gcd{\left(\binom M1,\binom M2,\binom M3,\ldots,\binom Mn\right)}=1$ where $M=\mathrm{lcm}(1,2,3,\ldots,n)$

Curious Binomial Coefficient Identity

Number of triangles inside given n-gon?

If a power of 2 divides a number, under what conditions does it divide a binomial coefficient involving the number that it divides?

Prove that $\prod_{k=1}^{\infty} \big\{(1+\frac1{k})^{k+\frac1{2}}\big/e\big\} = \dfrac{e}{\sqrt{2\pi}}$

Showing $\sum\limits_{j=0}^M \frac{M \choose j}{N+M \choose j} = \frac{N+M+1}{N+1}$

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

Odd Binomial Coefficients?

The binomial formula and the value of $0^0$

Dealing with a difficult sum of binomial coefficients, $\sum_{l=0}^{n}\binom{n}{l}^{2}\sum_{j=0}^{2l-n}\binom{l}{j} $

New posts in binomial-coefficients

Is there a closed form for $\sum_{n=1}^\infty\frac{(-4)^nH_{n-1}^3}{{2n\choose n}n^2}\ ?$

Sum of squares of binomial coefficients

Is $\sum_{k=1}^{n} k^{k-1} (n-k)^{2n-k} \binom{n}{k} \sim\frac{n^{2n}}{2\pi} $?

When is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $n,m,j$?

Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-

Coefficient extraction

Evaluating $\sum_{k=0}^n \binom{n}{k} 2^{k^2}$

Derive a closed form for a sum with inverse binomial coefficients

Intuition behind negative combinations

How do I prove this combinatorial identity using inclusion and exclusion principle?

Prove that $\gcd{\left(\binom M1,\binom M2,\binom M3,\ldots,\binom Mn\right)}=1$ where $M=\mathrm{lcm}(1,2,3,\ldots,n)$

Curious Binomial Coefficient Identity

Number of triangles inside given n-gon?

If a power of 2 divides a number, under what conditions does it divide a binomial coefficient involving the number that it divides?

Prove that $\prod_{k=1}^{\infty} \big\{(1+\frac1{k})^{k+\frac1{2}}\big/e\big\} = \dfrac{e}{\sqrt{2\pi}}$

Showing $\sum\limits_{j=0}^M \frac{M \choose j}{N+M \choose j} = \frac{N+M+1}{N+1}$

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

Odd Binomial Coefficients?

The binomial formula and the value of $0^0$

Dealing with a difficult sum of binomial coefficients, $\sum_{l=0}^{n}\binom{n}{l}^{2}\sum_{j=0}^{2l-n}\binom{l}{j} $