This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A011966 #42 Feb 04 2025 11:37:44 %S A011966 1,5,20,87,409,2066,11155,64077,389946,2504665,16923381,119928232, %T A011966 888980293,6876320041,55382419676,463539664643,4024626253845, %U A011966 36189297168874,336513491259647,3231446022478129,32004743929977258,326548129128737469,3428663026172389201 %N A011966 Third differences of Bell numbers. %C A011966 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Oct 29 2007 %D A011966 Olivier Gérard and Karol A. Penson, A budget of set partition statistics, in preparation, unpublished as of Sep 22 2011. %H A011966 Alois P. Heinz, <a href="/A011966/b011966.txt">Table of n, a(n) for n = 0..215</a> %H A011966 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 A011966 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 A011966 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 A011966 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 A011966 From _Vaclav Kotesovec_, Jul 28 2021: (Start) %F A011966 a(n) = Bell(n+3) - 3*Bell(n+2) + 3*Bell(n+1) - Bell(n). %F A011966 a(n) ~ n^3 * Bell(n) / LambertW(n)^3 * (1 - 3*LambertW(n)/n). (End) %p A011966 a:= n-> add((-1)^(k+1)*binomial(3,k)*combinat['bell'](n+k), k=0..3): %p A011966 seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 05 2008 %t A011966 Differences[BellB[Range[0,30]],3] (* _Harvey P. Dale_, Apr 21 2011 *) %o A011966 (Python) %o A011966 # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs. %o A011966 from itertools import accumulate %o A011966 A011966_list, blist, b = [1], [2, 3, 5], 5 %o A011966 for _ in range(1000): %o A011966 blist = list(accumulate([b]+blist)) %o A011966 b = blist[-1] %o A011966 A011966_list.append(blist[-4]) # _Chai Wah Wu_, Sep 20 2014 %Y A011966 Cf. A000110, A005493. A106436, A011965. %K A011966 nonn %O A011966 0,2 %A A011966 _N. J. A. Sloane_