

A011969


Apply (1+Shift)^2 to Bell numbers.


5



1, 3, 5, 10, 27, 87, 322, 1335, 6097, 30304, 162409, 931667, 5686712, 36750201, 250401793, 1792401626, 13436958559, 105208112643, 858286687914, 7279760687179, 64071719451645, 584150874832552, 5508179528996197
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

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 n1. 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 Gérard, 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, 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Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011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+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(n1)+Bell(n2).  Augustine O. Munagi, Jul 17 2008
G.f.: (1+2*x)*(1+x)^2*sum(k=>0 x^(2*k)*(4*x*k^22*k2*x1)/((2*k+1)*(1*x*k1))*A(k)/B(k) where A(k) = prod(p=0...k (2*p+1)), B(k) = prod(p=0...k (2*p1)*(2*x*px1)*(2*x*p2*x1)).  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*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, Jan 03 2013.


EXAMPLE

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.


MAPLE

with(combinat): 1, 3, seq(`if`(n>1, bell(n)+2*bell(n1)+bell(n2), NULL), n=2..22); # 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
A011969_list, blist, b, b2 = [1, 3], [1], 1, 1
for _ in range(10**2):
....blist = list(accumulate([b]+blist))
....A011969_list.append(2*b+b2+blist[1])
....b2, b = b, blist[1]
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014


CROSSREFS

Cf. A000110, A011968, A011970.
A diagonal of A011971 and A106436.  N. J. A. Sloane, Jul 31 2012
Sequence in context: A243517 A243518 A025486 * A003187 A262306 A284301
Adjacent sequences: A011966 A011967 A011968 * A011970 A011971 A011972


KEYWORD

nonn


AUTHOR

N. J. A. Sloane.


STATUS

approved



