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.
