%I #33 Apr 30 2019 21:50:00

%S 1,1,2,5,13,36,102,299,892,2713,8364,26108,82310,261804,838961,

%T 2706336,8780725,28636157,93818641,308641277,1019140129,3376604826,

%U 11221805968,37399728251,124967677989,418564867751,1405030366113,4726036692421,15927027834163,53770343259613

%N Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

%C Also the number of distinct orderless Mathematica expressions with one atom and n positions.

%H Andrew Howroyd, <a href="/A277996/b277996.txt">Table of n, a(n) for n = 1..200</a>

%H Mathematica Reference, <a href="http://reference.wolfram.com/mathematica/ref/Orderless.html">Orderless</a>.

%F From _Ilya Gutkovskiy_, Apr 30 2019: (Start)

%F G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} A(x^k)/k)).

%F G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + (Sum_{n>=1} a(n)*x^n) * Product_{n>=1} 1/(1 - x^n)^a(n)). (End)

%e The a(5)=13 Mathematica expressions are:

%e x[x,x,x]

%e x[x,x][] x[x][x] x[][x,x] x[x,x[]] x[x[x]]

%e x[x][][] x[][x][] x[][][x] x[x[]][] x[][x[]] x[x[][]]

%e x[][][][]

%t multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];

%t a[n_]:=a[n]=If[n===1,1,Sum[a[k]*Sum[Product[multing[a[First[s]],Length[s]],{s,Split[p]}],{p,IntegerPartitions[n-k-1]}],{k,1,n-1}]];

%t Array[a,30]

%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=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ _Andrew Howroyd_, Aug 19 2018

%Y Cf. A000108, A001003, A005043, A052893, A279944, A280000, A317658.

%K nonn

%O 1,3

%A _Gus Wiseman_, Dec 24 2016