The transformation 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 denotes the Iverson bracket and denotes the primes.
OEIS 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, ... A085507 0, 0, 1, 4, 13, 41, 136, 505, ... A024430 1, 0, 1, 3, 8, 25, 97, 434, 2095, ... A024429 0, 1, 1, 2, 7, 27, 106, 443, ... A033999 1, , 1, , 1, , ...Here, gives the Bell numbers.
has the exponential generating function
(3)
See alsoBinomial Transform,
Euler Transform,
Exponential Transform,
Möbius Transform,
Stirling Number of the First Kind,
Stirling Number of the Second Kind Explore with Wolfram|Alpha ReferencesBernstein, 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|AlphaStirling Transform Cite this as:Weisstein, Eric W. "Stirling Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StirlingTransform.html
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