Binomial Transform
المؤلف:
Bernstein, M. and Sloane, N. J. A.
المصدر:
"Some Canonical Sequences of Integers." Linear Algebra Appl. 226/228
الجزء والصفحة:
...
24-10-2020
1188
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.
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