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A360586 Expansion of e.g.f.: exp(x)*(exp(x)-1)*(exp(x)-x).

%I A360586 #10 Apr 02 2023 09:05:33
%S A360586 0,1,3,10,37,136,479,1618,5289,16876,52915,163846,502781,1532896,
%T A360586 4651911,14070394,42456913,127894996,384799067,1156756462,3475250085,
%U A360586 10436235976,31330727983,94038321250,282211432697,846835624636,2540926304259,7623651327958
%N A360586 Expansion of e.g.f.: exp(x)*(exp(x)-1)*(exp(x)-x).
%C A360586 a(n) is the number of ordered set partitions of an n-set into 3 sets such that the first set has at least one element, the second set cannot have a single element, and the third set has no restrictions.
%F A360586 a(n) = 3^n - 2^n - n*2^(n-1) + n.
%e A360586 The 37 set partitions for n=4 are the following:
%e A360586 {1,2,3,4}, {}, {}  (1 of these);
%e A360586 {1,2,3}, {}, {4}   (4 of this type);
%e A360586 {1,2}, {}, {3,4}   (6 of this type);
%e A360586 {1,2}, {3,4}, {}   (6 of this type);
%e A360586 {1}, {2,3}, {4}    (12 of this type);
%e A360586 {1}, {2,3,4}, {}   (4 of this type);
%e A360586 {1}, {}, {2,3,4}   (4 of this type).
%t A360586 With[{nn=30},CoefficientList[Series[Exp[x](Exp[x]-1)(Exp[x]-x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Apr 02 2023 *)
%K A360586 nonn
%O A360586 0,3
%A A360586 _Enrique Navarrete_, Feb 12 2023