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%I #77 Mar 10 2022 04:41:13
%S 1,2,7,27,114,523,2589,13744,77821,467767,2972432,19895813,139824045,
%T 1028804338,7905124379,63287544055,526827208698,4551453462543,
%U 40740750631417,377254241891064,3608700264369193,35613444194346451,362161573323083920,3790824599495473121
%N Second differences of Bell numbers.
%C Number of partitions of n+3 with at least one singleton and with the smallest element in a singleton equal to 3. Alternatively, number of partitions of n+3 with at least one singleton and with the largest element in a singleton equal to n+1. - Olivier GERARD, Oct 29 2007
%C Out of the A005493(n) set partitions with a specific two elements clustered separately, number that have a different set of two elements clustered separately. - Andrey Goder (andy.goder(AT)gmail.com), Dec 17 2007
%D Olivier Gérard and Karol A. Penson, A budget of set partition statistics, in preparation, unpublished as of Sep 22 2011.
%H Chai Wah Wu, <a href="/A011965/b011965.txt">Table of n, a(n) for n = 0..1000</a> n = 0..250 from Alois P. Heinz.
%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 Adam M. Goyt and Lara K. Pudwell, <a href="http://arxiv.org/abs/1203.3786">Avoiding colored partitions of two elements in the pattern sense</a>, arXiv preprint arXiv:1203.3786 [math.CO], 2012.
%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 a(n) = A005493(n) - A005493(n-1).
%F E.g.f.: exp(exp(x)-1)*(exp(2*x)-exp(x)+1). - _Vladeta Jovovic_, Feb 11 2003
%F a(n) = A000110(n) - 2*A000110(n-1) + A000110(n-2). - Andrey Goder (andy.goder(AT)gmail.com), Dec 17 2007
%F G.f.: G(0) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k+2*x-1) - x*(2*k+1)*(2*k+3)*(2*x*k+2*x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+3*x-1)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 19 2012
%F G.f.: 1 - G(0) where G(k) = 1 - 1/(1-k*x-2*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 17 2013
%F G.f.: 1 - 1/x + (1-x)^2/x/(G(0)-x) where G(k) = 1 - x*(k+1)/(1 - x/G(k+1) ); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 26 2013
%F G.f.: G(0)*(1-1/x) where G(k) = 1 - 1/(1-x*(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Feb 07 2013
%F a(n) ~ n^2 * Bell(n) / LambertW(n)^2 * (1 - 2*LambertW(n)/n). - _Vaclav Kotesovec_, Jul 28 2021
%F a(n) = Sum_{k=0..2^n - 1} b(k) for n >= 0 where b(2n+1) = b(n) + b(A025480(n-1)), b(2n) = b(n - 2^f(n)) + b(2n - 2^f(n)) + b(A025480(n-1)) for n > 0 with b(0) = b(1) = 1 and where f(n) = A007814(n). Also b((4^n - 1)/3) = A141154(n+1). - _Mikhail Kurkov_, Jan 27 2022
%p a:= n-> add((-1)^k*binomial(2,k)*combinat['bell'](n+k), k=0..2): seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 05 2008
%t Differences[BellB[Range[0, 30]], 2] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)
%o (Python)
%o # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
%o from itertools import accumulate
%o A011965_list, blist, b = [1], [1, 2], 2
%o for _ in range(1000):
%o ....blist = list(accumulate([b]+blist))
%o ....b = blist[-1]
%o ....A011965_list.append(blist[-3])
%o # _Chai Wah Wu_, Sep 02 2014
%Y Cf. A000110, A005493, A106436.
%Y Similar recurrences: A006014, A090365, A124758, A217924, A243499, A284005, A290615, A329369, A341392, A347204, A347205, A350309.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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