The binomial transform takes the sequence , , , ... to the sequence , , , ... via the transformation
The inverse transform is
(Sloane and Plouffe 1995, pp. 13 and 22). The inverse binomial transform of for prime and for composite is 0, 1, 3, 6, 11, 20, 37, 70, ... (OEIS A052467). The inverse binomial transform of for even and for odd is 0, 1, 2, 4, 8, 16, 32, 64, ... (OEIS A000079). Similarly, the inverse binomial transform of for odd and for even is 1, 2, 4, 8, 16, 32, 64, ... (OEIS A000079). The inverse binomial transform of the Bell numbers 1, 1, 2, 5, 15, 52, 203, ... (OEIS A000110) is a shifted version of the same numbers: 1, 2, 5, 15, 52, 203, ... (Bernstein and Sloane 1995, Sloane and Plouffe 1995, p. 22).
The central and raw moments of statistical distributions are also related by the binomial transform.