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%I #17 Jan 02 2020 08:24:28
%S 1,1,2,6,20,90,468,2910,20644,165874,1484344,14653890,158136988,
%T 1852077284,23394406084,317018563806,4587391330992,70598570456104,
%U 1151382852200680,19835976878704628,359963038816096924,6863033015330999110,137156667020252478684,2867083618970831936826
%N Number of balanced reduced multisystems with n atoms all equal to 1.
%C For n > 1, also the number of balanced reduced multisystems whose atoms are an integer partition of n with at least one part > 1. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. - _Gus Wiseman_, Dec 31 2019
%H Andrew Howroyd, <a href="/A318813/b318813.txt">Table of n, a(n) for n = 1..200</a>
%F a(n > 1) = A330679(n)/2. - _Gus Wiseman_, Dec 31 2019
%e The a(5) = 20 balanced reduced multisystems (with n written in place of 1^n):
%e 5 (14) (23) (113) (122) (1112)
%e ((1)(13)) ((1)(22)) ((1)(112))
%e ((3)(11)) ((2)(12)) ((2)(111))
%e ((11)(12))
%e ((1)(1)(12))
%e ((1)(2)(11))
%e (((1))((1)(12)))
%e (((1))((2)(11)))
%e (((2))((1)(11)))
%e (((12))((1)(1)))
%e (((11))((1)(2)))
%t normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t totfact[n_]:=totfact[n]=1+Sum[totfact[Times@@Prime/@normize[f]],{f,Select[facs[n],1<Length[#]<PrimeOmega[n]&]}];
%t Table[totfact[2^n],{n,10}]
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o seq(n)={my(v=vector(n, i, i==1), u=vector(n)); for(r=1, #v, u += v*sum(j=r, #v, (-1)^(j-r)*binomial(j-1, r-1)); v=EulerT(v)); u} \\ _Andrew Howroyd_, Dec 30 2019
%Y The maximum-depth case is A000111.
%Y Cf. A000311, A001055, A002846, A005121, A213427, A281118, A281119, A317145, A318812, A318846, A320154, A330474, A330679.
%K nonn
%O 1,3
%A _Gus Wiseman_, Sep 04 2018
%E Terms a(14) and beyond from _Andrew Howroyd_, Dec 30 2019
%E Terminology corrected by _Gus Wiseman_, Dec 31 2019
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