%I #36 Aug 04 2015 15:44:04
%S 1,4,8,15,37,114,409,1657,7432,36401,192713,1094076,6618379,42436913,
%T 287151994,2042803419,15229360185,118645071202,963494800557,
%U 8138047375093,71351480138824,648222594284197,6092330403828749
%N Apply (1+Shift)^3 to Bell numbers.
%C 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
%C 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
%D Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011
%H Chai Wah Wu, <a href="/A011970/b011970.txt">Table of n, a(n) for n = 0..500</a>
%H Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; <a href="http://www.jstor.org/stable/2310780">On the Number of Partitionings of a Set of n Distinct Objects</a>, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
%H Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; <a href="/A011965/a011965.pdf">On the Number of Partitionings of a Set of n Distinct Objects</a>, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
%H Augustine O. Munagi, <a href="http://dx.doi.org/10.1016/j.ejc.2007.06.003">Extended set partitions with successions</a>, European J. Combin. 29(5) (2008), 1298--1308.
%H Jocelyn Quaintance and Harris Kwong, <a href="http://www.emis.de/journals/INTEGERS/papers/n29/n29.Abstract.html">A combinatorial interpretation of the Catalan and Bell number difference tables</a>, Integers, 13 (2013), #A29.
%F If n>2, then bell(n)+3*bell(n-1)+3*bell(n-2)+bell(n-3). - _Augustine O. Munagi_, Jul 17 2008
%e a(3)=15 because the set {1,3,5,7} has 15 different partitions which are necessarily into blocks of nonconsecutive integers.
%p 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
%o (Python)
%o # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
%o from itertools import accumulate
%o A011970_list, blist, b, b2, b3 = [1,4,8], [1, 2], 2, 1, 1
%o for _ in range(498):
%o ....blist = list(accumulate([b]+blist))
%o ....A011970_list.append(3*(b+b2)+b3+blist[-1])
%o ....b3, b2, b = b2, b, blist[-1]
%o # _Chai Wah Wu_, Sep 02 2014, updated _Chai Wah Wu_, Sep 20 2014
%Y Cf. A000110.
%Y Cf. A011968, A011969.
%Y A diagonal of A011971 and A106436. - _N. J. A. Sloane_, Jul 31 2012
%K nonn
%O 0,2
%A _N. J. A. Sloane_.