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Divisiblity of binomial coefficients

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#number-theory #binomial-coefficients
96
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A proof in combinatorics (prove by any method)

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#combinatorics #proof-writing #binomial-coefficients
82
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If $P_r$ the coefficient of $x^r$ in the expansion of $(1+x)^2 \left(1+\frac{x}{2}\right)^2 \left(1+\frac{x}{2^2}\right)^2 \cdots$

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#calculus #binomial-coefficients
828
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Prove the following by two different methods, one combinatorial and one algebraic

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#combinatorics #proof-writing #binomial-coefficients
89
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Solving the following equation with imaginary roots

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#complex-numbers #binomial-coefficients #binomial-theorem
419
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Identity involving binomial coefficients

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#combinatorics #algebra-precalculus #modular-arithmetic #binomial-coefficients
82
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Why is this binomial equation not an identity between polynomials?

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#discrete-mathematics #binomial-coefficients
95
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Find $\sum_{k=0}^{n}4^k \binom{n}{k}$

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#combinatorics #binomial-coefficients
240
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odd property of Eulerian numbers

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#permutations #binomial-coefficients #stirling-numbers #eulerian-numbers
196
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prove the following algebraically $\left( \begin{array}{c} 2n \\ 2\ \end{array} \right) = 2 \left( \begin{array}{c} n \\ 2\ \end{array} \right) + n^2$

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#proof-verification #proof-writing #binomial-coefficients
188
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Another (complicated?) summation identity with binomial coefficients

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#combinatorics #binomial-coefficients
58
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Summation using Complex Numbers

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#complex-numbers #binomial-coefficients
48
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How to get $[z^n]\displaystyle\frac{z^k}{(1-z)^k} = {n-1 \choose k-1}?$

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#discrete-mathematics #binomial-coefficients #generating-functions
77
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Binomial Notation Expansion

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#binomial-coefficients #combinations
174
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Algebraically prove that $\sum\limits_{k=0}^n {n+k \choose n}\mathrm{2}^{n-k} = 1/2 \cdot \mathrm{2}^{2n+1}$

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#sequences-and-series #combinatorics #summation #binomial-coefficients

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