New posts in binomial-coefficients

Differentiating the binomial coefficient

real-analysis combinatorics derivatives binomial-coefficients

A conjecture formula: $\sum\limits_{n=1}^\infty \frac{\binom{mn}{n}}{n}\left(\frac{(m-1)^{m-1}}{m^m} \right)^n=m\log\left(\frac{m}{m-1}\right)$

power-series binomial-coefficients

Computing: $\lim\limits_{n\to\infty}\left(\prod\limits_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$

real-analysis limits binomial-coefficients

Binomials to polynomial

polynomials binomial-coefficients

Another summation identity with binomial coefficients

combinatorics binomial-coefficients

Sum of product of binomial coefficients: $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n + k}{k} = (-1)^n$

combinatorics summation binomial-coefficients

Prove by Combinatorial Argument that $\binom{n}{k}= \frac{n}{k} \binom{n-1}{k-1}$

combinatorics binomial-coefficients combinatorial-proofs

Why, conceptually, is it that $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$? [duplicate]

combinatorics binomial-coefficients intuition

Alternating sum of binomial coefficients multiplied by (1/k+1) [duplicate]

combinatorics summation binomial-coefficients

References on binomial coefficients of the form $\binom{-1/m}{k} $

combinatorics reference-request binomial-coefficients gamma-function

Evaluate $\sum_{k=0}^{n} {n \choose k}{m \choose k}$ for a given $n$ and $m$.

combinatorics summation binomial-coefficients

Combinatorial Identity $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$

combinatorics binomial-coefficients combinatorial-proofs

How can I express $\sum_{k=0}^n\binom{-1/2}{k}(-1)^k\binom{-1/2}{n-k}$ without using summations or minus signs?

binomial-coefficients

Prove that $2^n < \binom{2n}{n} < 2^{2n}$

combinatorics inequality binomial-coefficients

Prove $\sum_{k=0}^{58}\binom{2017+k}{58-k}\binom{2075-k}{k}=\sum_{k=0}^{29}\binom{4091-2k}{58-2k}$

combinatorics binomial-coefficients

Combinatorial proof of the identity $3^n=\sum_{k=0}^n \binom nk 2^k$

combinatorics summation binomial-coefficients combinatorial-proofs

How do you determine number of distinct integer solutions for an equation? [duplicate]

combinatorics discrete-mathematics binomial-coefficients diophantine-equations

Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$

combinatorics summation binomial-coefficients

Variation of Chu-Vandermonde binomial sum [duplicate]

sequences-and-series binomial-coefficients convolution

Binomial sum of $n$ terms in closed form

sequences-and-series binomial-coefficients summation