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%I #8 Oct 29 2023 20:26:56
%S 1,0,0,2,24,240,2440,26880,329280,4518080,69148800,1168675200,
%T 21564188800,430048819200,9195964377600,209593877292800,
%U 5068718054400000,129599032442880000,3492894468128665600,98968805893769011200,2940975338620999680000,91452266705317726208000,2969664371124258103296000
%N Expansion of e.g.f. exp(x^3/(3*(1-x)^3)).
%C 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).
%C 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.
%C A335344 is the corresponding sequence if 2 persons are selected from each table, and A000262 if only one person is selected from each table.
%e 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.
%t CoefficientList[Series[Exp[x^3/(3*(1-x)^3)],{x,0,22}],x]Table[n!,{n,0,22}] (* _Stefano Spezia_, Oct 02 2023 *)
%Y Cf. A000262, A001754, A335344.
%K nonn
%O 0,4
%A _Enrique Navarrete_, Oct 01 2023
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