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A011969 Apply (1+Shift)^2 to Bell numbers.

%I A011969 #79 Apr 04 2025 01:16:42
%S A011969 1,3,5,10,27,87,322,1335,6097,30304,162409,931667,5686712,36750201,
%T A011969 250401793,1792401626,13436958559,105208112643,858286687914,
%U A011969 7279760687179,64071719451645,584150874832552,5508179528996197
%N A011969 Apply (1+Shift)^2 to Bell numbers.
%C A011969 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
%C A011969 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
%D A011969 Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011
%H A011969 Chai Wah Wu, <a href="/A011969/b011969.txt">Table of n, a(n) for n = 0..500</a> n = 0..200 from Vincenzo Librandi.
%H A011969 Martin Cohn, Shimon Even, Karl Menger, Jr., and Philip K. Hooper, <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 A011969 Martin Cohn, Shimon Even, Karl Menger, Jr., and Philip K. Hooper, <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 A011969 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 A011969 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 A011969 For n >= 1, a(n+2) = exp(-1)*Sum_{k>=0} (k+1)^2/k!*k^n. - _Benoit Cloitre_, Mar 09 2008
%F A011969 If n>1, then a(n) = Bell(n) + 2*Bell(n-1) + Bell(n-2). - _Augustine O. Munagi_, Jul 17 2008
%F A011969 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)*(2*x*k-1))*A(k)/B(k) where A(k) = Product_{p=0..k} (2*p+1), B(k) = Product_{p=0..k} (2*p-1)*(2*x*p-x-1)*(2*x*p-2*x-1). - _Sergei N. Gladkovskii_, Jan 03 2013 [corrected by _Jason Yuen_, Apr 03 2025]
%F A011969 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) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 03 2013
%F A011969 a(n) ~ Bell(n) * (1 + 2*LambertW(n)/n). - _Vaclav Kotesovec_, Jul 28 2021
%e A011969 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.
%p A011969 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
%t A011969 Join[{1,3},#[[1]]+2#[[2]]+#[[3]]&/@Partition[BellB[Range[0,30]],3,1]] (* _Harvey P. Dale_, May 05 2023 *)
%o A011969 (Python)
%o A011969 # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
%o A011969 from itertools import accumulate
%o A011969 A011969_list, blist, b, b2 = [1,3], [1], 1, 1
%o A011969 for _ in range(10**2):
%o A011969     blist = list(accumulate([b]+blist))
%o A011969     A011969_list.append(2*b+b2+blist[-1])
%o A011969     b2, b = b, blist[-1]
%o A011969 # _Chai Wah Wu_, Sep 02 2014, updated _Chai Wah Wu_, Sep 20 2014
%Y A011969 Cf. A000110, A011968, A011970.
%Y A011969 A diagonal of A011971 and A106436. - _N. J. A. Sloane_, Jul 31 2012
%K A011969 nonn
%O A011969 0,2
%A A011969 _N. J. A. Sloane_