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# Complementary Bell Number

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

 (1)

where is a Stirling number of the second kind, are defined by analogy with the Bell numbers

 (2)

They are given by

 (3)

where is a Bell polynomial.

For , 1, ..., the first few are 1, , 0, 1, 1, , , , 50, 267, 413, ... (OEIS A000587).

They have generating function

 (4) (5) (6)

They have the series representation

 (7)

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

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 and ." 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 ." Fib. Quart. 7, 437-448, 1969.Yang, Y. "On a Multiplicative Partition Function." Electron. J. Combin. 8, No. R19, 2001.

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Complementary Bell Number

## Cite this as:

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