The Bell Series of Integers

Returns the values of the Bell series.

bell(n)

Arguments

n

Vector of non-negative integers. Values greater than 218 return an Inf. Non-integers or negative values return a NaN.

Details

The Bell numbers emerge from a series expansion of \( \exp(e^x - 1)\) for real \(x\). The first few values are \(B_{0}=1\), \(B_{1}=1\), \(B_{2}=2\), \(B_{3}=5\), \(B_{4}=15\). The series increases quickly so that overflow occurs when its argument is more than 218.

Value

This function returns \(B_{n}\).

References

Bell, E. T. (1934). Exponential polynomials. Ann. Math., 35, 258–277.

Bell, E. T. (1934). Exponential numbers. Amer. Math. Monthly, 41, 411–419.

Author

T. W. Yee

See also

bellff, rbell.

Examples

 if (FALSE) { # \dontrun{
plot(0:10, bell(0:10), log = "y", type = "h", col = "blue")
} # }