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How does one show that : $\int_{0}^{1}(\sin^{-1}x \cos^{-1}x)^n\mathrm dx=f(n)?$

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#calculus #integration #sequences-and-series #definite-integrals #binomial-coefficients
165
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Closed form for $S= \sum\limits_{k=0}^n x^k \binom{n}{k}^2$

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#combinatorics #summation #binomial-coefficients #binomial-theorem #legendre-polynomials
1.2k
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Find the sum of the coefficients of $x^{20}$ and $x^{21}$ in the power series expansion of $\frac 1{(1-x^3)^4}$.

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#power-series #binomial-coefficients #generating-functions
1.9k
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Evaluating $\sum_{r=0}^{10}\frac{\binom{10}{r}}{\binom{50}{30+r}}$

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#summation #binomial-coefficients
525
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Fibonacci Recursion with Binomial Coefficients

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#summation #binomial-coefficients #fibonacci-numbers
742
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Prove that $\binom{n}{k}=\binom{n}{k-1}\frac{n-k+1}{k}$

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#proof-writing #induction #binomial-coefficients #combinations
482
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Prove by induction the upper bound on the sum of binomial co-efficients

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#sequences-and-series #combinatorics #binomial-coefficients #machine-learning
225
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Evaluating Sum involving binomial coefficients and powers

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#sequences-and-series #combinatorics #binomial-coefficients #polylogarithm
173
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Showing two series are equal to each other

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#sequences-and-series #combinatorics #summation #binomial-coefficients #binomial-theorem
116
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Efficient evaluation of binomial series

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#numerical-methods #binomial-coefficients #binomial-theorem
430
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Pascal's triangle identity

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#discrete-mathematics #binomial-coefficients #combinations
114
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Given a progression 0, 1, 1, 2, 2, 2, 3, 3, 3, 3... What will be An?

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#binomial-coefficients #binomial-theorem #arithmetic-progressions
137
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Pascal's Triangle and Binomial Coefficients

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#binomial-coefficients
60
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Induction with Binomial Coefficients

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#combinatorics #induction #binomial-coefficients
106
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Sum with irrational powers and binomial coefficients

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#summation #binomial-coefficients

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