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The important binomial theorem states that sum_(k=0)^n(n; k)r^k=(1+r)^n. (1) Consider sums of powers of binomial coefficients a_n^((r)) = sum_(k=0)^(n)(n; k)^r (2) = ...

The binomial coefficient (n; k) is the number of ways of picking k unordered outcomes from n possibilities, also known as a combination or combinatorial number. The symbols ...

A binomial coefficient (N; k) is said to be exceptional if lpf(N; k)>N/k. The following table gives the exception binomial coefficients which are also good binomial ...

A binomial coefficient (N; k) with k>=2 is called good if its least prime factor satisfies lpf(N; k)>k (Erdős et al. 1993). This is equivalent to the requirement that GCD((N; ...

The q-binomial coefficient is a q-analog for the binomial coefficient, also called a Gaussian coefficient or a Gaussian polynomial. A q-binomial coefficient is given by [n; ...

The binomial distribution gives the discrete probability distribution P_p(n|N) of obtaining exactly n successes out of N Bernoulli trials (where the result of each Bernoulli ...

The converse of Fisher's theorem.

The binomial transform takes the sequence a_0, a_1, a_2, ... to the sequence b_0, b_1, b_2, ... via the transformation b_n=sum_(k=0)^n(-1)^(n-k)(n; k)a_k. The inverse ...

The nth central binomial coefficient is defined as (2n; n) = ((2n)!)/((n!)^2) (1) = (2^n(2n-1)!!)/(n!), (2) where (n; k) is a binomial coefficient, n! is a factorial, and n!! ...

A binomial number is a number of the form a^n+/-b^n, where a,b, and n are integers. Binomial numbers can be factored algebraically as ...