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A011970
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Apply (1+Shift)^3 to Bell numbers.
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4
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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
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OFFSET
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0,2
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COMMENTS
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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 Gérard, 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, Jul 17 2008
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REFERENCES
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Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011
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LINKS
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FORMULA
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EXAMPLE
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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
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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, Jul 17 2008
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PROG
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(Python)
# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011970_list, blist, b, b2, b3 = [1, 4, 8], [1, 2], 2, 1, 1
for _ in range(498):
....blist = list(accumulate([b]+blist))
....A011970_list.append(3*(b+b2)+b3+blist[-1])
....b3, b2, b = b2, b, blist[-1]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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