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The idempotent numbers are given by B_(n,k)(1,2,3,...)=(n; k)k^(n-k), where B_(n,k) is a Bell polynomial and (n; k) is a binomial coefficient. A table of the first few is ...

The Legendre transform of a sequence {c_k} is the sequence {a_k} with terms given by a_n = sum_(k=0)^(n)(n; k)(n+k; k)c_k (1) = sum_(k=0)^(n)(2k; k)(n+k; n-k)c_k, (2) where ...

x^n=sum_(k=0)^n<n; k>(x+k; n), where <n; k> is an Eulerian number and (n; k) is a binomial coefficient (Worpitzky 1883; Comtet 1974, p. 242).

The factorial n! is defined for a positive integer n as n!=n(n-1)...2·1. (1) So, for example, 4!=4·3·2·1=24. An older notation for the factorial was written (Mellin 1909; ...

Discrete Mathematics

A nexus number is a figurate number built up of the nexus of cells less than n steps away from a given cell. The nth d-dimensional nexus number is given by N_d(n) = ...

Possessing two modes. The term bimodal distribution, which refers to a distribution having two local maxima (as opposed to two equal most common values) is a slight ...

If x_1/n_1 and x_2/n_2 are the observed proportions from standard normally distributed samples with proportion of success theta, then the probability that ...

A smooth curve which corresponds to the limiting case of a histogram computed for a frequency distribution of a continuous distribution as the number of data points becomes ...

The Gaussian joint variable theorem, also called the multivariate theorem, states that given an even number of variates from a normal distribution with means all 0, (1) etc. ...