

A011968


Apply (1+Shift) to Bell numbers.


10



1, 2, 3, 7, 20, 67, 255, 1080, 5017, 25287, 137122, 794545, 4892167, 31858034, 218543759, 1573857867, 11863100692, 93345011951, 764941675963, 6514819011216, 57556900440429, 526593974392123, 4981585554604074, 48658721593531669, 490110875149889635
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OFFSET

0,2


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 Gérard, 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, Jul 17 2008


REFERENCES

Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011


LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..500 n = 0..200 from Vincenzo Librandi.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782785. MR1531841.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782785. MR1531841. [Annotated scanned copy]
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 12981308.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.


FORMULA

For n >= 1, a(n+1) = exp(1)*Sum_{k>=0} ((k+1)/k!)*k^n.  Benoit Cloitre, Mar 09 2008
For n >= 1, a(n) = Bell(n) + Bell(n1).  Augustine O. Munagi, Jul 17 2008
G.f.: G(0) where G(k) = 1  2*x*(k+1)/((2*k+1)*(2*x*k1)  x*(2*k+1)*(2*k+3)*(2*x*k1)/(x*(2*k+3)  2*(k+1)*(2*x*k+x1)/G(k+1) )); (recursively defined continued fraction).  Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1 + x*E(0) where E(k) = 1 + 1/(1x*kx)/(1x/(x+1/E(k+1) )); (recursively defined continued fraction).  Sergei N. Gladkovskii, Jan 20 2013
G.f.: 1 + Sum_{k>=0} ( 1+1/(1xx*k) )*x^(k+1)/Product_{i=0..k} (1x*i).  Sergei N. Gladkovskii, Jan 20 2013


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.


MAPLE

with(combinat): seq(`if`(n>0, bell(n)+bell(n1), 1), n=0..21); # Augustine O. Munagi, Jul 17 2008


PROG

(Python)
# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011968_list, blist, b = [1, 2], [1], 1
for _ in range(10**2):
....blist = list(accumulate([b]+blist))
....A011968_list.append(b+blist[1])
....b = blist[1] # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014


CROSSREFS

Cf. A000110, A000296, A011969, A011970.
A diagonal of A011971 and A106436.  N. J. A. Sloane, Jul 31 2012
Cf. A000569, A240936, A321750.
Sequence in context: A222867 A222776 A222890 * A080021 A306666 A032313
Adjacent sequences: A011965 A011966 A011967 * A011969 A011970 A011971


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



