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A366155
Expansion of e.g.f. exp(x^3/(3*(1-x)^3)).
0
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.
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
KEYWORD
nonn
AUTHOR
Enrique Navarrete, Oct 01 2023
STATUS
approved