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A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

%I A318813 #17 Jan 02 2020 08:24:28
%S A318813 1,1,2,6,20,90,468,2910,20644,165874,1484344,14653890,158136988,
%T A318813 1852077284,23394406084,317018563806,4587391330992,70598570456104,
%U A318813 1151382852200680,19835976878704628,359963038816096924,6863033015330999110,137156667020252478684,2867083618970831936826
%N A318813 Number of balanced reduced multisystems with n atoms all equal to 1.
%C A318813 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 A318813 Andrew Howroyd, <a href="/A318813/b318813.txt">Table of n, a(n) for n = 1..200</a>
%F A318813 a(n > 1) = A330679(n)/2. - _Gus Wiseman_, Dec 31 2019
%e A318813 The a(5) = 20 balanced reduced multisystems (with n written in place of 1^n):
%e A318813   5  (14)  (23)  (113)      (122)      (1112)
%e A318813                  ((1)(13))  ((1)(22))  ((1)(112))
%e A318813                  ((3)(11))  ((2)(12))  ((2)(111))
%e A318813                                        ((11)(12))
%e A318813                                        ((1)(1)(12))
%e A318813                                        ((1)(2)(11))
%e A318813                                        (((1))((1)(12)))
%e A318813                                        (((1))((2)(11)))
%e A318813                                        (((2))((1)(11)))
%e A318813                                        (((12))((1)(1)))
%e A318813                                        (((11))((1)(2)))
%t A318813 normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
%t A318813 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A318813 totfact[n_]:=totfact[n]=1+Sum[totfact[Times@@Prime/@normize[f]],{f,Select[facs[n],1<Length[#]<PrimeOmega[n]&]}];
%t A318813 Table[totfact[2^n],{n,10}]
%o A318813 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A318813 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 A318813 The maximum-depth case is A000111.
%Y A318813 Cf. A000311, A001055, A002846, A005121, A213427, A281118, A281119, A317145, A318812, A318846, A320154, A330474, A330679.
%K A318813 nonn
%O A318813 1,3
%A A318813 _Gus Wiseman_, Sep 04 2018
%E A318813 Terms a(14) and beyond from _Andrew Howroyd_, Dec 30 2019
%E A318813 Terminology corrected by _Gus Wiseman_, Dec 31 2019