OFFSET
2,2
COMMENTS
Previous name: Coefficients from fractional iteration of exp(x) - 1.
From Harry Richman, Mar 18 2023: (Start)
A "proper partition" of a set is a set partition in which there is more than one part, and there is some part which has more than one element.
a(n) is the number of chains of length 2 from the top element to the bottom element in the partition lattice on n labeled objects.
(End)
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 2..200
Ivo Rosenberg, The number of maximal closed classes in the set of functions over a finite domain, J. Combinatorial Theory Ser. A 14 (1973), 1-7. See Table I (it is not certain that this is the same sequence. - N. J. A. Sloane, Jun 25 2015)
Ivo Rosenberg and N. J. A. Sloane, Correspondence, 1971
FORMULA
a(n) = A000110(n) - 2.
EXAMPLE
For n = 3 there are a(3) = 3 proper partitions of {1,2,3}, which can be represented {12|3}, {13|2}, {23|1}.
For n = 4 there are a(4) = 13 proper partitions of {1,2,3,4}, which can be represented {123|4}, {124|3}, {134|2}, {234|1}, {12|34}, {13|24}, {14|23}, {12|3|4}, {13|2|4}, {14|2|3}, {23|1|4}, {24|1|3}, {34|1|2}.
MAPLE
seq(combinat[bell](n)-2, n=2..31); # Zerinvary Lajos, Sep 29 2006
MATHEMATICA
Table[BellB[n] - 2, {n, 3, 30}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
PROG
(PARI) a(n)=sum(j=2, n--, (j+1)*stirling(n, j, 2)) \\ Charles R Greathouse IV, Jul 06 2011
(Magma) [Bell(n) -2: n in [3..30]]; // G. C. Greubel, Sep 13 2019
(Sage) [bell_number(n)-2 for n in (3..30)] # G. C. Greubel, Sep 13 2019
(GAP) List([3..30], n-> Bell(n)-2); # G. C. Greubel, Sep 13 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Jan 02 2004
Name changed and a(2)=0 prepended by Harry Richman, Mar 18 2023
STATUS
approved