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Stirling Transform

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The Stirling transform S[{a_n}_(n=0)^N] of a sequence {a_n}_(n=0)^N is the sequence {b_n}_(n=0)^N defined by

b_n=sum_(k=0)^NS(n,k)a_k
(1)

for integers N>=0 and 0<=n<=N, where S(n,k) is a Stirling number of the second kind. The inverse transform is given by

a_n=sum_(k=0)^Ns(n,k)b_k,
(2)

where s(n,k) is a signed Stirling number of the first kind (Sloane and Plouffe 1995, p. 23).

The following table summarizes the beginnings of Stirling transforms for some common infinite sequences, where [S] denotes the Iverson bracket and P denotes the primes.

a_nOEISS[{a_n}_(n=0)^N]
1A0001101, 1, 2, 5, 15, 52, 203, ...
nA0054930, 1, 3, 10, 37, 151, 674, ...
n+1A0001101, 2, 5, 15, 52, 203, 877, ...
[n in P]A0855070, 0, 1, 4, 13, 41, 136, 505, ...
[n even]A0244301, 0, 1, 3, 8, 25, 97, 434, 2095, ...
[n odd]A0244290, 1, 1, 2, 7, 27, 106, 443, ...
(-1)^nn!A0339991, -1, 1, -1, 1, -1, ...

Here, S[{1}_(n=0)^N] gives the Bell numbers. The row for a_n=n is 0 followed by the terms of OEIS A005493, and the row for a_n=n+1 gives OEIS A000110 with its initial term omitted.

The Stirling transform of the infinite sequence a_n=n for n>=0 has the exponential generating function

g(x)=(e^x-1)exp(e^x-1).
(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. https://doi.org/10.1016/0024-3795(94)00245-9.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/M1484, A005493/M2851, A024429, A024430, A033999, 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.

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Stirling Transform

Cite this as:

Weisstein, Eric W. "Stirling Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StirlingTransform.html

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