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%I #12 Sep 01 2018 21:40:02
%S 1,0,1,0,2,2,5,10,18,46,94,212,476,1058,2441,5564,12880,29920,69620,
%T 163220,383376,904114,2139592,5074784,12074152,28789112,68803148,
%U 164779064,395373108,950416330,2288438591,5518864858,13329183894,32237132814,78069124640
%N Number of free pure identity multifunctions with one atom and n positions.
%C A free pure identity multifunction (PIM) 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 PIM, each of the g_i for i = 1, ..., k > 0 is a PIM, and for i != j we have g_i != g_j. The number of positions in a PIM is the number of brackets [...] plus the number of o's.
%H Andrew Howroyd, <a href="/A317877/b317877.txt">Table of n, a(n) for n = 1..200</a>
%e The a(8) = 10 PIMs:
%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]]]
%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 allIdPMF[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-2}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdPMF[h],Select[Tuples[allIdPMF/@p],UnsameQ@@#&]}],{p,Join@@Permutations/@IntegerPartitions[g]}]]];
%t Table[Length[allIdPMF[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]=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, A003238, A004111, A052893, A053492, A277996, A280000, A317875.
%Y Cf. A317876, A317878, A317879, A317880, A317881.
%K nonn
%O 1,5
%A _Gus Wiseman_, Aug 09 2018
%E Terms a(13) and beyond from _Andrew Howroyd_, Sep 01 2018
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