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A303022 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and no unitary parts (subexpressions of the form x[y]).

%I A303022 #16 Sep 11 2018 21:12:30
%S A303022 1,1,1,2,5,12,27,63,152,376,939,2371,6047,15577,40429,105637,277625,
%T A303022 733518,1947126,5190503,13888811,37291968,100444019,271316998,
%U A303022 734802247,1994873116,5427893149,14799525982,40429761365,110645688034,303316712450,832799212777
%N A303022 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and no unitary parts (subexpressions of the form x[y]).
%C A303022 Also the number of orderless Mathematica expressions with one atom, n positions, and no unitary parts.
%H A303022 Andrew Howroyd, <a href="/A303022/b303022.txt">Table of n, a(n) for n = 1..200</a>
%e A303022 The a(6) = 12 Mathematica expressions:
%e A303022   o[o,o[][]]
%e A303022   o[o[],o[]]
%e A303022   o[o,o,o[]]
%e A303022   o[o,o,o,o]
%e A303022   o[][o,o[]]
%e A303022   o[][o,o,o]
%e A303022   o[][][o,o]
%e A303022   o[o,o[]][]
%e A303022   o[o,o,o][]
%e A303022   o[][o,o][]
%e A303022   o[o,o][][]
%e A303022   o[][][][][]
%t A303022 allOLBF[n_]:=allOLBF[n]=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allOLBF[h],Select[Union[Sort/@Tuples[allOLBF/@p]],Length[#]!=1&]}],{p,IntegerPartitions[g]}]]];
%t A303022 Table[Length[allOLBF[n]],{n,10}]
%o A303022 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A303022 seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)-v); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ _Andrew Howroyd_, Aug 19 2018
%Y A303022 Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.
%Y A303022 Cf. A303023, A303024, A303025, A303026, A303027.
%K A303022 nonn
%O A303022 1,4
%A A303022 _Gus Wiseman_, Aug 15 2018
%E A303022 Terms a(21) and beyond from _Andrew Howroyd_, Aug 19 2018