A011970 Apply (1+Shift)^3 to Bell numbers.

1, 4, 8, 15, 37, 114, 409, 1657, 7432, 36401, 192713, 1094076, 6618379, 42436913, 287151994, 2042803419, 15229360185, 118645071202, 963494800557, 8138047375093, 71351480138824, 648222594284197, 6092330403828749

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

			a(3)=15 because the set {1,3,5,7} has 15 different partitions which are necessarily into blocks of nonconsecutive integers.
		

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• Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011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

Crossrefs

Cf. A000110.

Cf. A011968, A011969.

A diagonal of A011971 and A106436. - N. J. A. Sloane, Jul 31 2012

Programs

Maple

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

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]
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

Formula

If n>2, then bell(n)+3*bell(n-1)+3*bell(n-2)+bell(n-3). - Augustine O. Munagi, Jul 17 2008