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A011966 Third differences of Bell numbers. 5

%I

%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%D Olivier GĂ©rard and Karol A. Penson, A budget of set partition statistics, in preparation, unpublished as of Sep 22 2011.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

%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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Last modified June 5 18:48 EDT 2020. Contains 334854 sequences. (Running on oeis4.)