Complementary Bell Number

The complementary Bell numbers, also called the Uppuluri-Carpenter numbers,


where S(n,k) is a Stirling number of the second kind, are defined by analogy with the Bell numbers


They are given by


where B_n(x) is a Bell polynomial.

For n=0, 1, ..., the first few are 1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, ... (OEIS A000587).

They have generating function


They have the series representation


They are prime (in absolute value) for n=5, 36, 723, ... (OEIS A118018), corresponding to the prime numbers 2, 1454252568471818731501051, ... (OEIS A118019), with no others for n<=40968 (E. W. Weisstein, Mar. 21, 2009).

See also

Bell Number, Bell Polynomial, Integer Sequence Primes, Stirling Number of the Second Kind

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Complementary Bell Number

Cite this as:

Weisstein, Eric W. "Complementary Bell Number." From MathWorld--A Wolfram Web Resource.

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