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%I #18 Jan 03 2022 03:27:20
%S 1,2,6,24,119,702,4795,37183,322486,3091630,32453172,370104159,
%T 4555518746,60182704891,849245520581,12746759647944,202753756944382,
%U 3406596290534764,60282041591986049,1120554350714688128
%N Number of ways that n boxes with distinct sizes can contain each other under the condition that each box may contain at most three (themselves possibly nested) boxes. Each box is assumed to be large enough to contain any three smaller boxes.
%C If each box may contain at most one (possibly nested) box then the Bell numbers (A000110) are obtained, whereas if each box may contain at most two smaller (possibly nested) boxes then A000772 is obtained and if no restriction is placed on the number of (possible nested) boxes that any box may contain then the factorial numbers (A000142) are obtained. Sequence suggested by an earlier submission of _Rick L. Shepherd_.
%H Letong Hong and Rupert Li, <a href="https://arxiv.org/abs/2112.15081">Length-Four Pattern Avoidance in Inversion Sequences</a>, arXiv:2112.15081 [math.CO], 2021.
%F E.g.f.: exp(G(x) - 1), where G(x) be the function that satisfies 6G'(x) = G(x)^3 + 3G(x) + 2 and G(0) = 1. In this case G'(x) is the exponential generating function giving the number of ways to perform the given task if at most 3 boxes may fail to lie in another box. [_Joel B. Lewis_, Apr 28 2009]
%F a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x+x^2/2!+x^3/3!)*d/dx. Cf. A000110 and A000772. - Peter Bala, Nov 25 2011
%e a(3)=6, as seen from these arrangements: 112233, 321123, 311223, 211233, 223113, 113223, where xyyx indicates that box x contains box y, etc.
%t m = 30; G[_] = 1;
%t Do[G[x_] = 1 + (1/6) Integrate[G[x]^3 + 3 G[x] + 2, x] + O[x]^m, {m}];
%t CoefficientList[Exp[G[x] - 1] + O[x]^m, x] Range[0, m - 1]! // Rest (* _Jean-François Alcover_, Nov 13 2019 *)
%Y Cf. A000110, A000142, A000772.
%K nonn
%O 1,2
%A _John W. Layman_, May 25 2004
%E a(20) corrected by _Jean-François Alcover_, Nov 13 2019
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