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Complementary Bell Number

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The complementary Bell number B^~_n, also called an Uppuluri-Carpenter number, is defined for a nonnegative integer n by

B^~_n=sum_(k=0)^n(-1)^kS(n,k),
(1)

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

B_n=sum_(k=0)^nS(n,k).
(2)

They are given by

B^~_n=B_n(-1),
(3)

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).

Their exponential generating function G(x)=sum_(n=0)^(infty)B^~_nx^n/n! is

G(x)=ee^(-e^x)
(4)
=e^(1-e^x)
(5)
=1-x+1/6x^3+1/(24)x^4-1/(60)x^5-1/(80)x^6+....
(6)

They have the series representation

B^~_n=esum_(k=0)^infty((-1)^kk^n)/(k!),
(7)

where 0^0=1. 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, pers. comm., Mar. 21, 2009).

See also

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

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References

Beard, R. E. "On the Coefficients in the Expansion of e^(e^t) and e^(-e^t)." J. Inst. Actuaries 76, 152-163, 1950.Bouillet, J. E. "A Generalized Diffusion Equation: Radial Symmetries and Comparison Theorems." J. Math. Anal. Appl. 96, 37-68, 1983.Harris, B. and Schoenfeld, L. "Asymptotic Expansions for the Coefficients of Analytic Functions." Ill. J. Math. 12, 264-277, 1968.Klazar, M. "Counting Even and Odd Partitions." Amer. Math. Monthly 110, 527-532, 2003.Klazar, M. "Bell Numbers, Their Relatives, and Algebraic Differential Equations." J. Combin. Th. A 102, 63-87, 2003.Kolokolnikova, N. A. "Relations Between Sums of Certain Special Numbers." In Asymptotic and Enumeration Problems of Combinatorial Analysis (Ed. G. P. Egoryčev and M. L. Platonov). Krasnoyarsk, Soviet Union: Krasnojarsk. Gos. Univ., pp. 117-124, 1976.Sloane, N. J. A. Sequences A000587/M1913, A118018, and A118019 in "The On-Line Encyclopedia of Integer Sequences."Subbarao, M. V. and Verma, A. "Some Remarks on a Product Expansion. An Unexplored Partition Function." In Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999). Dordrecht, Netherlands: Kluwer, pp. 267-283, 2001.Uppuluri, V. R. R. and Carpenter, J. A. "Numbers Generated by the Function exp(1-e^x)." Fib. Quart. 7, 437-448, 1969.Yang, Y. "On a Multiplicative Partition Function." Elec. J. Combin. 8, No. 1, R19, 2001. https://doi.org/10.37236/1563.

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Complementary Bell Number

Cite this as:

Weisstein, Eric W. "Complementary Bell Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplementaryBellNumber.html

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