REDUCE

7.3 Bernoulli Numbers and Euler Numbers

The unary operator Bernoulli provides notation and computation for Bernoulli numbers. Bernoulli(n) evaluates to the nth Bernoulli number; all of the odd Bernoulli numbers, except Bernoulli(1), are zero.

The algorithms are based upon those by Herbert Wilf, presented by Sandra Fillebrown.[?]. If the ROUNDED switch is off, the algorithms are exactly those; if it is on, some further rounding may be done to prevent computation of redundant digits. Hence, these functions are particularly fast when used to approximate the Bernoulli numbers in rounded mode.

Euler numbers are computed by the unary operator Euler, which return the nth Euler number. The computation is derived directly from Pascal’s triangle of binomial coefficients.