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

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#calculus #algebra-precalculus #summation #binomial-coefficients
191
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Deriving $|\sum\limits_{k=0}^n (-1)^k \binom{n}{k}\binom{2n}{k}|\leq (\frac{16\sqrt 3}{9})^n $ where $|(z-1)^2(z+1)|\leq \frac{16\sqrt{3}}{9}$

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#complex-analysis #binomial-coefficients #binomial-theorem
2.7k
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Find the the coefficient of $\,x^r\,$ in $\,(1+x+x^2)^n$

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#sequences-and-series #taylor-expansion #binomial-coefficients #multinomial-theorem
77
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Proving Binomial Equivalence

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#combinatorics #discrete-mathematics #binomial-coefficients #binomial-theorem
111
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How to quickly find the $x^{24}$ term in this expansion?

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#binomial-coefficients
433
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Is row 14, column 4 for pascals triangle is 186?

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#abstract-algebra #binomial-coefficients
582
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How to solve a binomial inequality

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#probability #probability-theory #probability-distributions #binomial-coefficients #binomial-theorem
229
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Maclaurin Series by derivatives

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#calculus #taylor-expansion #binomial-coefficients
262
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Limit involving binomial coefficients without Stirling's formula

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#limits #binomial-coefficients
1.8k
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Probability of binomial n success before m failures?

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#probability #binomial-coefficients #negative-binomial
87
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Binomial sum involving power of $2$

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#binomial-coefficients
305
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Evaluate $\sum_{r=0}^{100} {(-1)^{r} {100 \choose r} {r^{50}}}$

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#combinatorics #discrete-mathematics #binomial-coefficients #binomial-theorem
176
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Is there a close form for $g(a,b,n)=\sum\limits_{k=0}^{n}\binom{n}{k}\frac{1}{ak+b}$?

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#summation #binomial-coefficients #factorial #egyptian-fractions
49
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Reduce two summations

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#summation #binomial-coefficients
296
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Get the probability, with Pascal's triangle.

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#probability #combinatorics #binomial-coefficients

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