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Prove using the method of double counting: $3^n =\sum_{k=0}^n \dbinom{n}{k} \sum_{j=0}^k \dbinom{k}{j}$ [duplicate]

Asymptotics of $\sum_{k=0}^{n} {\binom n k}^a$

Harmonic number identity

Inequality $\binom{2n}{n}\leq 4^n$

Prove that $ \sum_{k=0}^n \frac{(-1)^k}{k!}\binom{n}{k}=e\int_0^\infty \frac{t^ne^{-t}}{n!} J_0(2\sqrt{t})\;\mathrm{d}t$ using only real analysis.

Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$ [duplicate]

Proof of binomial coefficient formula.

Sum of the first integer powers of $n$ up to k

Techniques for summing ratio of binomial coefficients

Probability of winning a prize in a raffle

Show $\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$ by block-walking interpretation of Pascal's triangle

Prove $\sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}.$

How to prove $\sum_{s=0}^{m}{2s\choose s}{s\choose m-s}\frac{(-1)^s}{s+1}=(-1)^m$?

Evaluation of ratio of two binomial expression

Combinatorial Proof of $\binom{\binom{n}{2}}{2} = 3 \binom{n}{3}+ 3 \binom{n}{4}$ for $n \geq 4$

Inverse of the $n$-by-$n$ matrix $(a_{jk})$ where $a_{jk} = \binom{j-1}{k-1}$

New posts in binomial-coefficients

Some trouble with the induction

Is there a closed form for $\sum_{n=1}^\infty\frac{2^{2n}H_n}{n^3{2n\choose n}}?$

How can I prove, that this formula is related to the binomial series?

How can I prove the identity $2(n-1)n^{n-2}=\sum_k\binom{n}{k}k^{k-1}(n-k)^{n-k-1}$?

Prove using the method of double counting: $3^n =\sum_{k=0}^n \dbinom{n}{k} \sum_{j=0}^k \dbinom{k}{j}$ [duplicate]

Asymptotics of $\sum_{k=0}^{n} {\binom n k}^a$

Harmonic number identity

Inequality $\binom{2n}{n}\leq 4^n$

Prove that $ \sum_{k=0}^n \frac{(-1)^k}{k!}\binom{n}{k}=e\int_0^\infty \frac{t^ne^{-t}}{n!} J_0(2\sqrt{t})\;\mathrm{d}t$ using only real analysis.

Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$ [duplicate]

Proof of binomial coefficient formula.

Sum of the first integer powers of $n$ up to k

Techniques for summing ratio of binomial coefficients

Probability of winning a prize in a raffle

Show $\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$ by block-walking interpretation of Pascal's triangle

Prove $\sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}.$

How to prove $\sum_{s=0}^{m}{2s\choose s}{s\choose m-s}\frac{(-1)^s}{s+1}=(-1)^m$?

Evaluation of ratio of two binomial expression

Combinatorial Proof of $\binom{\binom{n}{2}}{2} = 3 \binom{n}{3}+ 3 \binom{n}{4}$ for $n \geq 4$

Inverse of the $n$-by-$n$ matrix $(a_{jk})$ where $a_{jk} = \binom{j-1}{k-1}$