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A011970
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Apply (1+Shift)^3 to Bell numbers.
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0
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1, 4, 8, 15, 37, 114, 409, 1657, 7432, 36401, 192713, 1094076, 6618379, 42436913, 287151994, 2042803419, 15229360185, 118645071202, 963494800557, 8138047375093, 71351480138824, 648222594284197, 6092330403828749
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Starting with n=3 (a(3)=15), number of set partitions of n+2 with at least one singleton and the smallest element in any singleton is exactly n-2. The maximum number of singletons is therefore 5. Alternatively, starting with n=3, number of set partitions of n+2 with at least one singleton and the largest element in any singleton is exactly 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 3 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>2, 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| Olivier Gerard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011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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FORMULA
| If n>2, then bell(n)+3*bell(n-1)+3*bell(n-2)+bell(n-3). - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008
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EXAMPLE
| a(3)=15 because the set {1,3,5,7} has 15 different partitions which are necessarily into blocks of nonconsecutive integers.
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MAPLE
| with(combinat): 1, 4, 8, seq(`if`(n>2, bell(n)+3*bell(n-1)+3*bell(n-2)+bell(n-3), NULL), n=3..22); - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008
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CROSSREFS
| Cf. A000110.
Cf. A011968, A011969.
Sequence in context: A027961 A018921 A103536 * A111988 A110652 A059373
Adjacent sequences: A011967 A011968 A011969 * A011971 A011972 A011973
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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