The transformation S[{a_n}_(n = 0)^N] of a sequence {a_n}_(n = 0)^N into a sequence {b_n}_(n = 0)^N by the formula b_n = sum_(k = 0)^N S(n, k) a_k, where S(n, k) is a Stirling number of the second kind. The inverse transform is given by a_n = sum_(k = 0)^N s(n, k) b_k, where s(n, k) is a Stirling number of the first kind. The following table summarized Stirling transforms for some common sequences, where [S] denotes the Iverson bracket and P denotes the primes.

binomial transform | Euler transform | exponential transform | Möbius transform | Stirling number of the first kind | Stirling number of the second kind