Stirling Transform
المؤلف:
Bernstein, M. and Sloane, N. J. A.
المصدر:
"Some Canonical Sequences of Integers." Linear Algebra Appl. 226-228
الجزء والصفحة:
...
5-11-2020
2373
Stirling Transform
The transformation ![S[<span style=]()
{a_n}_(n=0)^N]" src="https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline1.gif" style="height:21px; width:65px" /> of a sequence ![<span style=]()
{a_n}_(n=0)^N" src="https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline2.gif" style="height:19px; width:42px" /> into a sequence ![<span style=]()
{b_n}_(n=0)^N" src="https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline3.gif" style="height:19px; width:42px" /> by the formula
 |
(1)
|
where 
is a Stirling number of the second kind. The inverse transform is given by
 |
(2)
|
where 
is a Stirling number of the first kind (Sloane and Plouffe 1995, p. 23).
The following table summarized Stirling transforms for some common sequences, where ![[S]](https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline6.gif)
denotes the Iverson bracket and 
denotes the primes.
 |
OEIS |
![S[<span style=]() {a_n}_(n=0)^N]" src="https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline9.gif" style="height:21px; width:65px" /> |
| 1 |
A000110 |
1, 1, 2, 5, 15, 52, 203, ... |
 |
A005493 |
0, 1, 3, 10, 37, 151, 674, ... |
 |
A000110 |
1, 2, 5, 15, 52, 203, 877, ... |
![[n in P]](https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline12.gif) |
A085507 |
0, 0, 1, 4, 13, 41, 136, 505, ... |
![[n even]](https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline13.gif) |
A024430 |
1, 0, 1, 3, 8, 25, 97, 434, 2095, ... |
![[n odd]](https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline14.gif) |
A024429 |
0, 1, 1, 2, 7, 27, 106, 443, ... |
 |
A033999 |
1,  , 1,  , 1,  , ... |
Here, ![S[<span style=]()
{1}_(n=0)^N]" src="https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline19.gif" style="height:21px; width:59px" /> gives the Bell numbers.
![S[<span style=]()
{n}_(n=0)^N]" src="https://mathworld.wolfram.com/images/equations/StirlingTransform/Inline20.gif" style="height:21px; width:59px" /> has the exponential generating function
 |
(3)
|
REFERENCES:
Bernstein, M. and Sloane, N. J. A. "Some Canonical Sequences of Integers." Linear Algebra Appl. 226-228, 57-72, 1995.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Factorial Factors." §4.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 252, 1994.
Riordan, J. Combinatorial Identities. New York: Wiley, p. 90, 1979.
Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 48, 1980.
Sloane, N. J. A. Sequences A000110/M1483, A005493/M2851, A024429, A024430, A033999, A052437, and A085507 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.
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
