%I #8 Aug 19 2018 17:23:30

%S 1,0,1,0,2,1,5,5,15,23,54,98,212,420,886,1822,3838,8046,17029,36097,

%T 76889,164245,351971,756341,1629389,3518643,7614717,16512962,35875986,

%U 78082171,170219300,371651968,812624721,1779240627,3900634491,8561723769,18814112811

%N Number of free pure symmetric identity multifunctions with one atom and n positions.

%C A free pure symmetric identity multifunction (SIM) is either (case 1) the leaf symbol "o", or (case 2) an expression of the form h[g_1, ..., g_k] where h is a SIM, each of the g_i for i = 1, ..., k > 0 is a SIM, and for i < j we have g_i < g_j under a canonical total ordering such as the Mathematica ordering of expressions. The number of positions in a SIM is the number of brackets [...] plus the number of o's.

%H Andrew Howroyd, <a href="/A317878/b317878.txt">Table of n, a(n) for n = 1..200</a>

%e The a(8) = 5 SIMs:

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

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

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

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

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

%t allIdPMFOL[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-2}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdPMFOL[h],Select[Union[Sort/@Tuples[allIdPMFOL/@p]],UnsameQ@@#&]}],{p,IntegerPartitions[g]}]]];

%t Table[Length[allIdPMFOL[n]],{n,12}]

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

%o seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)); v=concat(v, sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ _Andrew Howroyd_, Aug 19 2018

%Y Cf. A000081, A003238, A004111, A052893, A053492, A277996, A280000, A317875.

%Y Cf. A317876, A317877, A317879, A317880, A317881.

%K nonn

%O 1,5

%A _Gus Wiseman_, Aug 09 2018

%E Terms a(13) and beyond from _Andrew Howroyd_, Aug 19 2018