OFFSET
0,3
COMMENTS
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
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]
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..574 (terms n=0..200 from Vincenzo Librandi)
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.
FORMULA
E.g.f: x*exp(x)*exp(exp(x)-1).
Sum_{k=1..n} n*binomial(n-1, k-1)*Bell(n-k), n >= 2. - Zerinvary Lajos, Nov 22 2006
a(n) ~ n^(n+1) * exp(n/LambertW(n)-1-n) / (sqrt(1+LambertW(n)) * LambertW(n)^n). - Vaclav Kotesovec, Mar 13 2014
a(n) = Sum_{k=1..n} k * A175757(n,k). - Alois P. Heinz, Mar 03 2020
a(n) = Sum_{j=0..n} n * Stirling2(n,j). - Detlef Meya, Apr 11 2024
MAPLE
with(combinat): a:=n->sum(numbcomb (n, 0)*bell(n), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Apr 25 2007
with(combinat): a:=n->sum(bell(n), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Apr 25 2007
a:=n->sum(sum(Stirling2(n, k), j=1..n), k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jun 28 2007
MATHEMATICA
a[n_] := n!*Coefficient[Series[x E^(E^x+x-1), {x, 0, n}], x, n]
Table[Sum[BellB[n, 1], {i, 1, n}], {n, 0, 21}] (* Zerinvary Lajos, Jul 16 2009 *)
Table[n*BellB[n], {n, 0, 20}] (* Vaclav Kotesovec, Mar 13 2014 *)
PROG
(PARI) a(n)=local(t); if(n<0, 0, t=exp(x+O(x^n)); n!*polcoeff(x*t*exp(t-1), n))
(Sage) [bell_number(n)*n for n in range(22) ] # Zerinvary Lajos, Mar 14 2009
(Magma) [n*Bell(n): n in [0..25]]; // Vincenzo Librandi, Mar 15 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Apr 19 2002
STATUS
approved