A366155 Expansion of e.g.f. exp(x^3/(3*(1-x)^3)).

1, 0, 0, 2, 24, 240, 2440, 26880, 329280, 4518080, 69148800, 1168675200, 21564188800, 430048819200, 9195964377600, 209593877292800, 5068718054400000, 129599032442880000, 3492894468128665600, 98968805893769011200, 2940975338620999680000, 91452266705317726208000, 2969664371124258103296000

Offset

0,4

Comments

For n>0, a(n) is the number of ways to split n people into nonempty groups, have each group sit around a circular table, and select 3 people from each table (where two seating arrangements are considered identical if each person has the same left neighbors in both of them).

2*A001754(n) is the number of ways to seat n persons around a circular table and select 3 of them if only one table is used.

A335344 is the corresponding sequence if 2 persons are selected from each table, and A000262 if only one person is selected from each table.

Links

Table of n, a(n) for n=0..22.

Example

a(7)=26880 since, using one table, there are 6! circular seatings and binomial(7,3) ways to select 3 persons, hence 25200 ways. Using two tables, the only way we can select 3 persons from each one is seating 4 persons in one table and 3 in the other, which can be done in 420 ways; then choosing 3 persons from each table can be done in 4 ways, for a total of 1680 ways; hence 25200 + 1680 = 26880.

Mathematica

CoefficientList[Series[Exp[x^3/(3*(1-x)^3)], {x, 0, 22}], x]Table[n!, {n, 0, 22}] (* Stefano Spezia, Oct 02 2023 *)

Crossrefs

Cf. A000262, A001754, A335344.

Sequence in context: A025131 A270564 A143407 * A228619 A252764 A215929

Adjacent sequences: A366152 A366153 A366154 * A366156 A366157 A366158

Keyword

nonn

Author

Enrique Navarrete, Oct 01 2023

Status

approved