The transformation ![S[{a_n}_(n=0)^N]](/images/equations/StirlingTransform/Inline1.svg)
of a sequence 
into a sequence 
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]](/images/equations/StirlingTransform/Inline6.svg)
denotes the Iverson bracket and 
denotes the primes.
 | OEIS | ![S[{a_n}_(n=0)^N]](/images/equations/StirlingTransform/Inline9.svg) |
| 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]](/images/equations/StirlingTransform/Inline12.svg) | A085507 | 0, 0, 1, 4, 13, 41, 136, 505, ... |
![[n even]](/images/equations/StirlingTransform/Inline13.svg) | A024430 | 1, 0, 1, 3, 8, 25, 97, 434, 2095, ... |
![[n odd]](/images/equations/StirlingTransform/Inline14.svg) | A024429 | 0, 1, 1, 2, 7, 27, 106, 443, ... |
 | A033999 | 1,  ,
1,  ,
1,  ,
... |
Here, ![S[{1}_(n=0)^N]](/images/equations/StirlingTransform/Inline19.svg)
gives the Bell numbers.
![S[{n}_(n=0)^N]](/images/equations/StirlingTransform/Inline20.svg)
has the exponential generating function
 |
(3)
|
See also
Binomial Transform,
Euler Transform,
Exponential Transform,
Möbius Transform,
Stirling
Number of the First Kind,
Stirling
Number of the Second Kind
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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.Referenced
on Wolfram|Alpha
Stirling Transform
Cite this as:
Weisstein, Eric W. "Stirling Transform."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StirlingTransform.html
Subject classifications
