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A320266 Number of balanced orderless tree-factorizations of n.

%I A320266 #8 Nov 18 2018 15:00:03
%S A320266 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,6,1,4,1,4,2,2,1,8,2,2,3,4,1,5,1,9,2,2,
%T A320266 2,11,1,2,2,8,1,5,1,4,4,2,1,17,2,4,2,4,1,8,2,8,2,2,1,13,1,2,4,19,2,5,
%U A320266 1,4,2,5,1,24,1,2,4,4,2,5,1,17,6,2,1,13,2
%N A320266 Number of balanced orderless tree-factorizations of n.
%C A320266 A rooted tree is balanced if all leaves are the same distance from the root.
%C A320266 An orderless tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless tree-factorizations, one of each factor in a factorization of n.
%C A320266 a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Nov 18 2018
%H A320266 Andrew Howroyd, <a href="/A320266/b320266.txt">Table of n, a(n) for n = 1..10000</a>
%F A320266 a(p^n) = A320160(n) for prime p. - _Andrew Howroyd_, Nov 18 2018
%e A320266 The a(36) = 11 balanced orderless tree-factorizations:
%e A320266   36,
%e A320266   (2*18), (3*12), (4*9), (6*6),
%e A320266   (2*2*9), (2*3*6), (3*3*4),
%e A320266   (2*2*3*3), ((2*2)*(3*3)), ((2*3)*(2*3)).
%t A320266 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A320266 oltfacs[n_]:=If[n<=1,{{}},Prepend[Union@@Function[q,Sort/@Tuples[oltfacs/@q]]/@DeleteCases[facs[n],{n}],n]];
%t A320266 Table[Length[Select[oltfacs[n],SameQ@@Length/@Position[#,_Integer]&]],{n,100}]
%o A320266 (PARI) MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
%o A320266 seq(n)={my(u=vector(n, i, 1), v=vector(n)); while(u, v+=u; u[1]=1; u=MultEulerT(u)-u); v} \\ _Andrew Howroyd_, Nov 18 2018
%Y A320266 Cf. A000669, A001055, A048816, A050336, A281118, A292505, A119262, A120803, A141268, A292504, A319312, A320160, A320267.
%K A320266 nonn
%O A320266 1,4
%A A320266 _Gus Wiseman_, Oct 08 2018