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A303027 Number of free pure symmetric multifunctions with one atom, n positions, and no empty or unitary parts (subexpressions of the form x[] or x[y]).

%I A303027 #16 Sep 11 2018 21:15:11
%S A303027 1,0,0,1,1,1,3,5,7,15,28,47,90,175,319,607,1181,2251,4325,8449,16425,
%T A303027 31992,62823,123521,243047,480316,951290,1886293,3749341,7467815,
%U A303027 14893500,29752398,59532947,119274491,239275400,480638121,966571853,1945901716,3921699524
%N A303027 Number of free pure symmetric multifunctions with one atom, n positions, and no empty or unitary parts (subexpressions of the form x[] or x[y]).
%C A303027 Also the number of orderless Mathematica expressions with one atom, n positions, and no empty or unitary parts.
%H A303027 Andrew Howroyd, <a href="/A303027/b303027.txt">Table of n, a(n) for n = 1..200</a>
%e A303027 The a(10) = 15 Mathematica expressions:
%e A303027   o[o,o[o,o[o,o]]]
%e A303027   o[o,o[o,o][o,o]]
%e A303027   o[o[o,o],o[o,o]]
%e A303027   o[o,o][o,o[o,o]]
%e A303027   o[o,o[o,o]][o,o]
%e A303027   o[o,o][o,o][o,o]
%e A303027   o[o,o[o,o,o,o,o]]
%e A303027   o[o,o,o[o,o,o,o]]
%e A303027   o[o,o,o,o[o,o,o]]
%e A303027   o[o,o,o,o,o[o,o]]
%e A303027   o[o,o][o,o,o,o,o]
%e A303027   o[o,o,o][o,o,o,o]
%e A303027   o[o,o,o,o][o,o,o]
%e A303027   o[o,o,o,o,o][o,o]
%e A303027   o[o,o,o,o,o,o,o,o]
%t A303027 allOLZR[n_]:=allOLZR[n]=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allOLZR[h],Select[Union[Sort/@Tuples[allOLZR/@p]],Length[#]>1&]}],{p,IntegerPartitions[g]}]]];
%t A303027 Table[Length[allOLZR[n]],{n,25}]
%o A303027 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A303027 seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)-v); v=concat(v, sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ _Andrew Howroyd_, Aug 19 2018
%Y A303027 Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.
%Y A303027 Cf. A303022, A303023, A303024, A303025, A303026.
%K A303027 nonn
%O A303027 1,7
%A A303027 _Gus Wiseman_, Aug 15 2018
%E A303027 Terms a(29) and beyond from _Andrew Howroyd_, Aug 19 2018