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 the prime number indicator 
(1 for prime 
and 0 otherwise) begins 0, 0, 1, 4, 10,
21, 41, 78, .... Its first differences are
0, 1, 3, 6, 11, 20, 37, 70, ... (OEIS A052467).
The inverse binomial transform of 
for even 
and 
for odd 
is 1, 1, 2, 4, 8, 16, 32, 64, ... (
for 
, giving OEIS A000079).
Similarly, the inverse binomial transform of 
for odd 
and 
for even 
is 0, 1, 2, 4, 8, 16, 32, 64, ... (
for 
, giving 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 moments and raw moments of statistical distributions are related by expanding about the mean using the binomial theorem when the relevant moments exist.
