|
%I #56 Apr 11 2024 17:21:15
%S 0,1,4,15,60,260,1218,6139,33120,190323,1159750,7464270,50563164,
%T 359377681,2672590508,20744378175,167682274352,1408702786668,
%U 12277382510862,110822101896083,1034483164707440,9972266139291771,99147746245841106,1015496134666939958
%N a(n) = n*B(n), where B(n) are the Bell numbers, A000110.
%C a(n) is the total number of successions among all partitions of {1,2,...,n+1}; a succession is a pair (i,i+1) of consecutive integers lying in a block. For example, a(3)=15 because {1,2,3,4} has 6 partitions with 1 succession - 1/2/34, 1/23/4, 12/3/4, 14/23, 134/2, 124/3, 3 partitions with 2 successions - 1/234, 123/4, 12/34 and 1 partition with 3 successions - 1234. Thus a(3) = 6*1 + 3*2 + 1*3 = 15. - _Augustine O. Munagi_, Jul 01 2008
%C a(n) is the number of occurrences of integers in a list of all partitions of the set {1,...,n}. For example, the list 123, 1/23, 2/13, 3/12, 1/2/3 of all partitions of the set {1,2,3} requires 15 occurrences of integers each belonging to that set. [From Michael Hardy (hardy(AT)math.umn.edu), Nov 08 2008]
%C The bijection between the two foregoing characterizations is as follows: Fix x in {1,2,...,n} and associate x with the succession (x,x+1) which appears in some partitions of {1,2,...,n+1}. Replace x,x+1 by x and partition the n-set {1,2,...,x,x+2,...,n+1}, giving B(n) partitions. Thus the succession (x,x+1) occurs among partitions of {1,2,...,n+1} exactly B(n) times. - _Augustine O. Munagi_, Jun 02 2010
%H Alois P. Heinz, <a href="/A070071/b070071.txt">Table of n, a(n) for n = 0..574</a> (terms n=0..200 from Vincenzo Librandi)
%H Augustine O. Munagi, <a href="http://dx.doi.org/10.1016/j.ejc.2007.06.003">Extended set partitions with successions</a>, European J. Combin. 29(5) (2008), 1298--1308.
%F E.g.f: x*exp(x)*exp(exp(x)-1).
%F Sum_{k=1..n} n*binomial(n-1, k-1)*Bell(n-k), n >= 2. - _Zerinvary Lajos_, Nov 22 2006
%F a(n) ~ n^(n+1) * exp(n/LambertW(n)-1-n) / (sqrt(1+LambertW(n)) * LambertW(n)^n). - _Vaclav Kotesovec_, Mar 13 2014
%F a(n) = Sum_{k=1..n} k * A175757(n,k). - _Alois P. Heinz_, Mar 03 2020
%F a(n) = Sum_{j=0..n} n * Stirling2(n,j). - _Detlef Meya_, Apr 11 2024
%p with(combinat): a:=n->sum(numbcomb (n,0)*bell(n), j=1..n): seq(a(n), n=0..21); # _Zerinvary Lajos_, Apr 25 2007
%p with(combinat): a:=n->sum(bell(n), j=1..n): seq(a(n), n=0..21); # _Zerinvary Lajos_, Apr 25 2007
%p a:=n->sum(sum(Stirling2(n, k), j=1..n), k=1..n): seq(a(n), n=0..21); # _Zerinvary Lajos_, Jun 28 2007
%t a[n_] := n!*Coefficient[Series[x E^(E^x+x-1), {x, 0, n}], x, n]
%t Table[Sum[BellB[n, 1], {i, 1, n}], {n, 0, 21}] (* _Zerinvary Lajos_, Jul 16 2009 *)
%t Table[n*BellB[n], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 13 2014 *)
%o (PARI) a(n)=local(t); if(n<0,0,t=exp(x+O(x^n)); n!*polcoeff(x*t*exp(t-1),n))
%o (Sage) [bell_number(n)*n for n in range(22) ] # _Zerinvary Lajos_, Mar 14 2009
%o (Magma) [n*Bell(n): n in [0..25]]; // _Vincenzo Librandi_, Mar 15 2014
%Y Cf. A000110, A052889, A105479, A105480, A105481, A175757.
%Y Row sums of A270236, A270701, A270702, A286416, A319298, A319375.
%K nonn,changed
%O 0,3
%A _Karol A. Penson_, Apr 19 2002
|
