Search: id:A011968
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%I A011968
%S A011968 1,2,3,7,20,67,255,1080,5017,25287,137122,794545,4892167,
%T A011968 31858034,218543759,1573857867,11863100692,93345011951,
%U A011968 764941675963,6514819011216,57556900440429,526593974392123
%N A011968 Apply (1+Shift) to Bell numbers.
%C A011968 Number of set partitions of n+2 with at least one singleton and the smallest 
               element in any singleton is exactly n. The maximum number of singletons 
               is therefore 3. Alternatively, number of set partitions of n+2 with 
               at least one singleton and the largest element in any singleton is 
               exactly 3 (or n+2 if n+2<3). For example, a(3)=7 counts the following 
               set partitions of [5]: {1245, 3}, {12, 3, 45}, {124, 3, 5}, {15, 
               24, 3}, {125, 3, 4}, {14, 25, 3}, {12, 3, 4, 5} - Olivier GERARD 
               (olivier.gerard(AT)gmail.com), Oct 29 2007
%C A011968 Let V(N)={v(1),v(2),...,v(N)} denote an ordered set of increasing positive 
               integers containing a pair of adjacent elements that differ by at 
               least 2, that is, v(i),v(i+1) with v(i+1)-v(i)>1. Then for n>0, a(n) 
               is the number of partitions of V(n+1) into blocks of nonconsecutive 
               integers. - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008
%D A011968 Olivier Gerard and Karol Penson, A budget of set partitions statistics, 
               in preparation.
%D A011968 A. O. Munagi, Extended set partitions with successions, European J. Combin. 
               29(5) (2008), 1298--1308.
%F A011968 for n>=1 a(n+1)= exp(-1)*sum(k>=0,(k+1)/k!*k^n) - Benoit Cloitre (abmt(AT)wanadoo.fr), 
               Mar 09 2008
%F A011968 If n>0, then a(n)=bell(n)+bell(n-1). - Augustine O. Munagi (amunagi(AT)yahoo.com), 
               Jul 17 2008
%e A011968 a(3)=7 because the set {1,3,4,5} has 7 different partitions into blocks 
               of nonconsecutive integers: 14/35, 135/4, 1/35/4, 13/4/5, 14/3/5, 
               15/3/4, 1/3/4/5.
%p A011968 with(combinat): seq(`if`(n>0,bell(n)+bell(n-1),1),n=0..21); - Augustine 
               O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008
%Y A011968 Cf. A000110.
%Y A011968 Cf. A000296.
%Y A011968 Sequence in context: A110498 A006073 A052402 this_sequence A080021 A032313 
               A032223
%Y A011968 Adjacent sequences: A011965 A011966 A011967 this_sequence A011969 A011970 
               A011971
%K A011968 nonn
%O A011968 0,2
%A A011968 N. J. A. Sloane (njas(AT)research.att.com).

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