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 A070071 a(n) = n*B(n), where B(n) are the Bell numbers, A000110. 13
 0, 1, 4, 15, 60, 260, 1218, 6139, 33120, 190323, 1159750, 7464270, 50563164, 359377681, 2672590508, 20744378175, 167682274352, 1408702786668, 12277382510862, 110822101896083, 1034483164707440, 9972266139291771, 99147746245841106, 1015496134666939958 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000110, A052889, A105479, A105480, A105481, A175757. Row sums of A270236, A270701, A270702, A286416, A319298, A319375. Sequence in context: A291244 A290910 A369838 * A285363 A356942 A151484 Adjacent sequences: A070068 A070069 A070070 * A070072 A070073 A070074 KEYWORD nonn AUTHOR Karol A. Penson, Apr 19 2002 STATUS approved

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