%I #20 Sep 12 2018 12:46:23

%S 1,1,2,4,11,29,83,251,767,2403,7652,24758,80875,266803,887330,2972108,

%T 10016981,33942461,115572864,395226810,1356840007,4674552089,

%U 16156355357,56003840659,194651585875,678220460687,2368505647624,8288873657180,29064904732911

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

%C A free pure identity multifunction (with empty expressions allowed) (IME) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where h is an IME, each of the g_i for i = 1, ..., k >= 0 is an IME, and for i != j we have g_i != g_j. The number of positions in an IME is the number of brackets [...] plus the number of o's.

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

%H Andrew Howroyd, <a href="/A317879/b317879.txt">Table of n, a(n) for n = 1..500</a>

%e The a(5) = 11 IMEs:

%e o[o[o]]

%e o[o][o]

%e o[o[][]]

%e o[o[],o]

%e o[o,o[]]

%e o[][o[]]

%e o[][][o]

%e o[o[]][]

%e o[][o][]

%e o[o][][]

%e o[][][][]

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

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

%o (PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, my(p=prod(k=1, n, 1 + sum(i=1, n\k, binomial(v[k], i)*x^(i*k)*y^i) + O(x*x^n))); v[n]=v[n-1]+sum(k=1, n-2, v[n-k-1]*subst(serlaplace(y^0*polcoef(p, k)), y, 1))); v} \\ _Andrew Howroyd_, Sep 01 2018

%Y Cf. A000081, A001003, A004111, A277996, A280000, A317875.

%Y Cf. A317876, A317877, A317878, A317880, A317881.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 09 2018

%E Terms a(13) and beyond from _Andrew Howroyd_, Sep 01 2018