%I #16 Sep 11 2018 21:15:11

%S 1,0,0,1,1,1,3,5,7,15,28,47,90,175,319,607,1181,2251,4325,8449,16425,

%T 31992,62823,123521,243047,480316,951290,1886293,3749341,7467815,

%U 14893500,29752398,59532947,119274491,239275400,480638121,966571853,1945901716,3921699524

%N 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 Also the number of orderless Mathematica expressions with one atom, n positions, and no empty or unitary parts.

%H Andrew Howroyd, <a href="/A303027/b303027.txt">Table of n, a(n) for n = 1..200</a>

%e The a(10) = 15 Mathematica expressions:

%e o[o,o[o,o[o,o]]]

%e o[o,o[o,o][o,o]]

%e o[o[o,o],o[o,o]]

%e o[o,o][o,o[o,o]]

%e o[o,o[o,o]][o,o]

%e o[o,o][o,o][o,o]

%e o[o,o[o,o,o,o,o]]

%e o[o,o,o[o,o,o,o]]

%e o[o,o,o,o[o,o,o]]

%e o[o,o,o,o,o[o,o]]

%e o[o,o][o,o,o,o,o]

%e o[o,o,o][o,o,o,o]

%e o[o,o,o,o][o,o,o]

%e o[o,o,o,o,o][o,o]

%e o[o,o,o,o,o,o,o,o]

%t 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 Table[Length[allOLZR[n]],{n,25}]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o 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 Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.

%Y Cf. A303022, A303023, A303024, A303025, A303026.

%K nonn

%O 1,7

%A _Gus Wiseman_, Aug 15 2018

%E Terms a(29) and beyond from _Andrew Howroyd_, Aug 19 2018