Idempotent Number
The idempotent numbers are given by
where 
is a Bell polynomial and 
is a binomial coefficient. A table of the first few is given below.
| |
 |
 |
 |
 |
 |
 |
 |
 |
A000027 |
A001788 |
A036216 |
A040075 |
A050982 |
A050988 |
A050989 |
| 1 |
1 |
|
|
|
|
|
|
| 2 |
2 |
1 |
|
|
|
|
|
| 3 |
3 |
6 |
1 |
|
|
|
|
| 4 |
4 |
24 |
12 |
1 |
|
|
|
| 5 |
5 |
80 |
90 |
20 |
1 |
|
|
| 6 |
6 |
240 |
540 |
240 |
30 |
1 |
|
| 7 |
7 |
672 |
2835 |
2240 |
525 |
42 |
1 |
| 8 |
8 |
1792 |
13608 |
17920 |
7000 |
1008 |
56 |
| 9 |
9 |
4608 |
61236 |
129024 |
78750 |
18144 |
1764 |
| 10 |
10 |
11520 |
262440 |
860160 |
787500 |
272160 |
41160 |
REFERENCES:
Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 91, 1974.
Roman, S. The Umbral Calculus. New York: Academic Press, p. 85, 1984.
Sloane, N. J. A. Sequences A000027/M0472, A001788/M4161, A036216, A040075, A050982, A050988, and A050989 in "The On-Line Encyclopedia of Integer Sequences."
