Search: id:A011969
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%I A011969
%S A011969 1,3,5,10,27,87,322,1335,6097,30304,162409,931667,5686712,36750201,
%T A011969 250401793,1792401626,13436958559,105208112643,858286687914,
%U A011969 7279760687179,64071719451645,584150874832552,5508179528996197
%N A011969 Apply (1+Shift)^2 to Bell numbers.
%C A011969 Starting with n=2 (a(2)=5), number of set partitions of n+2 with at least 
               one singleton and the smallest element in any singleton is exactly 
               n-1. The maximum number of singletons is therefore 4. Alternatively, 
               starting with n=2, number of set partitions of n+2 with at least 
               one singleton and the largest element in any singleton is exactly 
               4. E.g. a(3)=10 counts the following set partitions of [5]: {1345, 
               2}, {13, 2, 45}, {145, 2, 3}, {134, 2, 5}, {15, 2, 34}, {135, 2, 
               4}, {14, 2, 35}, {13, 2, 4, 5}, {14, 2, 3, 5}, {15, 2, 3, 4} - Olivier 
               GERARD (olivier.gerard(AT)gmail.com), Oct 29 2007
%C A011969 Let V(N)={v(1),v(2),...,v(N)} denote an ordered set of increasing positive 
               integers containing 2 pairs 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>1, 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 A011969 Olivier Gerard and Karol Penson, A budget of set partitions statistics, 
               in preparation.
%D A011969 A. O. Munagi, Extended set partitions with successions, European J. Combin. 
               29(5) (2008), 1298--1308.
%F A011969 For n>=1, a(n+2)= exp(-1)*sum(k>=0,(k+1)^2/k!*k^n) - Benoit Cloitre (abmt(AT)wanadoo.fr), 
               Mar 09 2008
%F A011969 If n>1, then a(n)=bell(n)+2*bell(n-1)+bell(n-2) - Augustine O. Munagi 
               (amunagi(AT)yahoo.com), Jul 17 2008
%e A011969 a(3)=10 because the set {1,3,5,6} has 10 different partitions into blocks 
               of nonconsecutive integers: 15/36, 16/35, 135/6, 136/5, 1/35/6, 1/
               36/5, 13/5/6, 15/3/6, 16/3/5, 1/3/5/6.
%p A011969 with(combinat): 1,3,seq(`if`(n>1,bell(n)+2*bell(n-1)+bell(n-2),NULL),
               n=2..22); - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008
%Y A011969 Cf. A000110.
%Y A011969 Cf. A011968.
%Y A011969 Sequence in context: A132332 A002039 A007695 this_sequence A003187 A100885 
               A003186
%Y A011969 Adjacent sequences: A011966 A011967 A011968 this_sequence A011970 A011971 
               A011972
%K A011969 nonn
%O A011969 0,2
%A A011969 N. J. A. Sloane (njas(AT)research.att.com).

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