Binomial Transform

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.

REFERENCES:

Bernstein, M. and Sloane, N. J. A. "Some Canonical Sequences of Integers." Linear Algebra Appl. 226/228, 57-72, 1995.

Sloane, N. J. A. Sequences A000079/M1129, A000110/M1484, and A052467 in "The On-Line Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.