cp's OEIS Frontend

A094198 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.

%I A094198 #18 Jan 03 2022 03:27:20
%S A094198 1,2,6,24,119,702,4795,37183,322486,3091630,32453172,370104159,
%T A094198 4555518746,60182704891,849245520581,12746759647944,202753756944382,
%U A094198 3406596290534764,60282041591986049,1120554350714688128
%N A094198 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 A094198 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 A094198 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 A094198 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 A094198 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 A094198 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 A094198 m = 30; G[_] = 1;
%t A094198 Do[G[x_] = 1 + (1/6) Integrate[G[x]^3 + 3 G[x] + 2, x] + O[x]^m, {m}];
%t A094198 CoefficientList[Exp[G[x] - 1] + O[x]^m, x] Range[0, m - 1]! // Rest (* _Jean-François Alcover_, Nov 13 2019 *)
%Y A094198 Cf. A000110, A000142, A000772.
%K A094198 nonn
%O A094198 1,2
%A A094198 _John W. Layman_, May 25 2004
%E A094198 a(20) corrected by _Jean-François Alcover_, Nov 13 2019