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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 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 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Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011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, 782--785. 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, 782--785. MR1531841. [Annotated scanned copy]

Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.

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(n-1)+Bell(n-2). - Augustine O. Munagi, 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.

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(n-1)+bell(n-2), 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: A243516 A243517 A243518 * A003187 A262306 A284301

Adjacent sequences:  A011966 A011967 A011968 * A011970 A011971 A011972

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 15 04:00 EST 2018. Contains 317225 sequences. (Running on oeis4.)