Newbetuts

Proving that $\sum_{k=0}^{2n} {2k \choose k } {2n \choose k}\left( \frac{-1}{2} \right)^k=4^{-n}~{2n \choose n}.$

Binomial expansion for $(x+a)^n$ for non-integer n

$2^\sqrt{10}$ vs $3^2$

How to find closed form of a binomial series.

Properties of Sigma sum

Mistake in expanding binomial theorem

Prove that for $(1-x)^m (1+x)^n$ there is no $k$ such that the coefficients of $x^k$ and $x^{k+1}$ are both $0$

Smoothstep sigmoid-like function: Can anyone prove this relation?

Find the coefficient of the term $x^2$ in $\left(x+\frac 2x\right)^{4}$

Binomial theorem proof for rational index without calculus

Show that $\lim [(2n)^\frac{1}{n})] = 1$

Binomial theorem for matrices

Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem

Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $ [duplicate]

What is the proof of the Binomial Theorem, other than the induction method? How can we find the expansion of binomails with indices like 2n, 3n, 4n..?

New posts in binomial-theorem

Why does the Binomial Theorem use combinations and not permutations for its coefficients?

Coefficient of $x^{n-2}$ in $(x-1)(x-2)(x-3)\dotsm(x-n)$

Prove that $16 ^ {2023} + 1$ is divisible by $17 ^ 2$.

Sum with binomial coefficients: $\sum_{m=1}^{k}\frac{1}{m^{2a}}\binom{k}{m}$ with constant a

Finding the last two digits of a number by binomial theorem

Proving that $\sum_{k=0}^{2n} {2k \choose k } {2n \choose k}\left( \frac{-1}{2} \right)^k=4^{-n}~{2n \choose n}.$

Binomial expansion for $(x+a)^n$ for non-integer n

$2^\sqrt{10}$ vs $3^2$

How to find closed form of a binomial series.

Properties of Sigma sum

Mistake in expanding binomial theorem

Prove that for $(1-x)^m (1+x)^n$ there is no $k$ such that the coefficients of $x^k$ and $x^{k+1}$ are both $0$

Smoothstep sigmoid-like function: Can anyone prove this relation?

Find the coefficient of the term $x^2$ in $\left(x+\frac 2x\right)^{4}$

Binomial theorem proof for rational index without calculus

Show that $\lim [(2n)^\frac{1}{n})] = 1$

Binomial theorem for matrices

Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem

Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $ [duplicate]

What is the proof of the Binomial Theorem, other than the induction method? How can we find the expansion of binomails with indices like 2n, 3n, 4n..?