Stirling Transform

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)

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.