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%I #22 Jan 01 2024 11:30:29
%S 1,1,2,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Number of ways to write n! as product of distinct squarefree numbers.
%C From _Gus Wiseman_, Aug 23 2020: (Start)
%C Also the number of set-systems (sets of sets) whose multiset union is the multiset of prime factors of n!. For example, the a(1) = 1 through a(7) = 3 set-systems (empty columns indicated by dots) are:
%C 0 {1} {1,2} . {1},{1,2},{1,3} . {1},{1,2},{1,3},{1,2,4}
%C {1},{2} {1},{1,2},{1,4},{1,2,3}
%C {1},{2},{1,2},{1,3},{1,4}
%C (End)
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F a(n) = 0 for n > 7;
%F a(n) = A050326(A000142(n)).
%e n=7, 7! = 1*2*3*4*5*6*7 = 5040 = 2*2*2*2*3*3*5*7: a(7) = #{2*3*6*10*14, 2*6*10*42, 2*6*14*30} = 3.
%t yst[n_]:=yst[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[yst[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t Table[Length[yst[n!]],{n,15}] (* _Gus Wiseman_, Aug 21 2020 *)
%Y A103774 is the non-strict version.
%Y A337073 is the version for superprimorials, with non-strict version A337072.
%Y A001055 counts factorizations.
%Y A045778 counts strict factorizations.
%Y A048656 counts squarefree divisors of factorials.
%Y A050320 counts factorizations into squarefree numbers.
%Y A050326 counts strict factorizations into squarefree numbers.
%Y A050342 counts set-systems by total sum.
%Y A076716 counts factorizations of factorials.
%Y A116539 counts set-systems covering an initial interval.
%Y A157612 counts strict factorizations of factorials.
%Y Cf. A000110, A005117, A008480, A089259, A116540, A124010, A318360.
%Y Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.
%K nonn
%O 1,3
%A _Reinhard Zumkeller_, Feb 15 2005
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