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A011968
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Apply (1+Shift) to Bell numbers.
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6
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1, 2, 3, 7, 20, 67, 255, 1080, 5017, 25287, 137122, 794545, 4892167, 31858034, 218543759, 1573857867, 11863100692, 93345011951, 764941675963, 6514819011216, 57556900440429, 526593974392123
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
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
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REFERENCES
| Olivier Gerard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011
A. O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
| 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
If n>0, then a(n)=bell(n)+bell(n-1). - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008
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EXAMPLE
| 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.
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MAPLE
| 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
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CROSSREFS
| Cf. A000110.
Cf. A000296.
Sequence in context: A171655 A006073 A052402 * A080021 A032313 A032223
Adjacent sequences: A011965 A011966 A011967 * A011969 A011970 A011971
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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