login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 4
1, 2, 6, 24, 119, 702, 4795, 37183, 322486, 3091630, 32453172, 370104159, 4555518746, 60182704891, 849245520581, 12746759647944, 202753756944382, 3406596290534764, 60282041591986049, 1120554350714688128 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
Letong Hong and Rupert Li, Length-Four Pattern Avoidance in Inversion Sequences, arXiv:2112.15081 [math.CO], 2021.
FORMULA
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]
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
EXAMPLE
a(3)=6, as seen from these arrangements: 112233, 321123, 311223, 211233, 223113, 113223, where xyyx indicates that box x contains box y, etc.
MATHEMATICA
m = 30; G[_] = 1;
Do[G[x_] = 1 + (1/6) Integrate[G[x]^3 + 3 G[x] + 2, x] + O[x]^m, {m}];
CoefficientList[Exp[G[x] - 1] + O[x]^m, x] Range[0, m - 1]! // Rest (* Jean-François Alcover, Nov 13 2019 *)
CROSSREFS
Sequence in context: A256207 A256208 A264432 * A297200 A071077 A202213
KEYWORD
nonn
AUTHOR
John W. Layman, May 25 2004
EXTENSIONS
a(20) corrected by Jean-François Alcover, Nov 13 2019
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)