New posts in binomial-coefficients

Show that ${{2m} \choose {m}} \leq \frac{2^{2m}}{\sqrt{2m}}$.

combinatorics inequality binomial-coefficients

An upper bound for $\sum_{i = 1}^m \binom{i}{k}\frac{1}{2^i}$?

combinatorics binomial-coefficients

Prove that $\sum\limits_{k=0}^{m}\binom{m}{k}\binom{n+k}{m}=\sum\limits_{k=0}^{m}\binom{n}{k}\binom{m}{k}2^k$ [duplicate]

combinatorics summation binomial-coefficients

How to find the sum $\sum_{k=1}^{\lfloor n/2\rfloor}\frac{2^{n-2k}\binom{n-2}{2k-2}\binom{2k-2}{k-1}}{k}$

combinatorics summation contest-math binomial-coefficients catalan-numbers

A Curious Binomial Sum Identity without Calculus of Finite Differences

combinatorics polynomials binomial-coefficients interpolation

The integral $\int_0^1 \frac{(x+1)^n-1}{x} dx$

binomial-coefficients definite-integrals

How to compute the asymptotic growth of $\binom{n}{\log n}$?

asymptotics logarithms binomial-coefficients

Find the coefficient of $x^3y^2z^3$ in the expansion $(2x+3y-4z+w)^9$

algebra-precalculus binomial-coefficients

Binomial analogue of Riemann sum for definite integral

real-analysis calculus number-theory limits binomial-coefficients

Simplify $\sum \limits_{k=0}^{n} \binom{n}{k} 2^{\sqrt{k}}$

combinatorics binomial-coefficients

Is it possible to derive the nth derivative of$~\exp\left(x\right)\sin^{}\left(x\right)~$using binomial coefficient$~{n\choose k}~$?

derivatives trigonometry summation systems-of-equations binomial-coefficients

$\sum\limits_{k=0}^{\frac{p-1}2}3^k\binom{p}{k}\equiv 2^p-1\pmod{p^2}$

elementary-number-theory contest-math binomial-coefficients

Closed form expression for the harmonic sum $\sum\limits_{n=1}^{\infty}\frac{H_{2n}}{n^2\cdot4^n}{2n \choose n}$

real-analysis sequences-and-series definite-integrals binomial-coefficients harmonic-numbers

How to evaluate $1 - \frac{\binom{n^2}{1}}{\binom{n+1}{1}} + \frac{\binom{n^2}{2}}{\binom{n+2}{2}} - \frac{\binom{n^2}{3}}{\binom{n+3}{3}} + ..$

combinatorics binomial-coefficients

Can this be proved using Combinatorics or generating functions?

combinatorics summation binomial-coefficients generating-functions combinatorial-proofs

Are these two binomial sums known? Proven generalization to the Hockey Stick patterns in Pascal's Triangle

combinatorics summation binomial-coefficients

How can I derive what is $1\cdot 2\cdot 3\cdot 4 + 2\cdot 3\cdot 4\cdot 5+ 3\cdot 4\cdot 5\cdot 6+\cdots + (n-3)(n-2)(n-1)(n)$ ??

calculus sequences-and-series summation proof-writing binomial-coefficients

How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$

sequences-and-series combinatorics summation binomial-coefficients

Find the value $\binom {n}{0} + \binom{n}{4} + \binom{n}{8} + \cdots $, where $n$ is a positive integer.

summation power-series binomial-coefficients

Sum of reciprocals of binomial coefficients: $ \sum\limits_{k=0}^{n-1}\frac1{\binom{n}{k}(n-k)} $

combinatorics summation binomial-coefficients