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%I #37 Dec 27 2021 22:09:14
%S 1,5,20,87,409,2066,11155,64077,389946,2504665,16923381,119928232,
%T 888980293,6876320041,55382419676,463539664643,4024626253845,
%U 36189297168874,336513491259647,3231446022478129,32004743929977258,326548129128737469,3428663026172389201
%N Third differences of Bell numbers.
%C Number of partitions of n+4 with at least one singleton and with the smallest element in a singleton equal to 4. Alternatively, number of partitions of n+4 with at least one singleton and with the largest element in a singleton equal to n+1. - Olivier GERARD, Oct 29 2007
%D Olivier Gérard and Karol A. Penson, A budget of set partition statistics, in preparation, unpublished as of Sep 22 2011.
%H Alois P. Heinz, <a href="/A011966/b011966.txt">Table of n, a(n) for n = 0..215</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 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 G.f.: -(1-x+x^2)/x^2 + (1-x)^3/x^2/(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 From _Vaclav Kotesovec_, Jul 28 2021: (Start)
%F a(n) = Bell(n+3) - 3*Bell(n+2) + 3*Bell(n+1) - Bell(n).
%F a(n) ~ n^3 * Bell(n) / LambertW(n)^3 * (1 - 3*LambertW(n)/n). (End)
%p a:= n-> add ((-1)^(k+1) *binomial(3,k) *combinat['bell'](n+k), k=0..3): seq (a(n), n=0..20); # _Alois P. Heinz_, Sep 05 2008
%t Differences[BellB[Range[0,30]],3] (* _Harvey P. Dale_, Apr 21 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 A011966_list, blist, b = [1], [2, 3, 5], 5
%o for _ in range(1000):
%o ....blist = list(accumulate([b]+blist))
%o ....b = blist[-1]
%o ....A011966_list.append(blist[-4]) # _Chai Wah Wu_, Sep 20 2014
%Y Cf. A000110, A005493. A106436, A011965.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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