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A011969
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Apply (1+Shift)^2 to Bell numbers.
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5
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1, 3, 5, 10, 27, 87, 322, 1335, 6097, 30304, 162409, 931667, 5686712, 36750201, 250401793, 1792401626, 13436958559, 105208112643, 858286687914, 7279760687179, 64071719451645, 584150874832552, 5508179528996197
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OFFSET
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0,2
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COMMENTS
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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
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
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REFERENCES
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Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; Mathematical Notes: On the Number of Partitionings of a Set of n Distinct Objects. Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841 - From N. J. A. Sloane, Jul 31 2012
Olivier Gerard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011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
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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For n>=1, a(n+2)= exp(-1)*sum(k>=0,(k+1)^2/k!*k^n) - Benoit Cloitre, Mar 09 2008
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
G.f.: -(1+2*x)*(1+x)^2*sum(k=>0 x^(2*k)*(4*x*k^2-2*k-2*x-1)/((2*k+1)*(1*x*k-1))*A(k)/B(k) where A(k) = prod(p=0...k (2*p+1)), B(k) = prod(p=0...k (2*p-1)*(2*x*p-x-1)*(2*x*p-2*x-1)) . - Sergei N. Gladkovskii, Jan 03 2013.
G.f.: G(0)*(1+x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 03 2013.
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EXAMPLE
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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.
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MAPLE
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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
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CROSSREFS
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Cf. A000110, A011968, A011970.
A diagonal of A011971. - N. J. A. Sloane, Jul 31 2012
Sequence in context: A209001 A171867 A002039 * A003187 A003186 A006826
Adjacent sequences: A011966 A011967 A011968 * A011970 A011971 A011972
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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