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%I #29 Jan 01 2021 18:12:02
%S 1,1,3,11,43,187,872,4375,23258,130485,767348,4710715,30070205,
%T 198983975,1361361925,9607908808,69812787049,521377973359,
%U 3996036977270,31389624598631,252408597286705,2075472033455894,17434190966525003,149476993511444023,1307022313790487959
%N Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions.
%C A free pure symmetric multifunction (with empty expressions allowed) f in EOME is either (case 1) a positive integer, or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where k >= 0, h is in EOME, each of the g_i for i = 1, ..., k is in EOME, and for i < j we have g_i <= g_j under a canonical total ordering of EOME, such as the Mathematica ordering of expressions.
%C Also the number of inequivalent colorings of orderless Mathematica expressions with n positions.
%e Inequivalent representatives of the a(3) = 11 colorings:
%e 1[1,1] 1[2,2] 1[1,2] 1[2,3]
%e 1[1[]] 1[2[]]
%e 1[][1] 1[][2]
%e 1[1][] 1[2][]
%e 1[][][]
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
%o InequivalentColoringsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Dec 30 2020
%Y Row sums of A304485.
%Y Cf. A000612, A007716, A052893, A053492, A277996, A279944, A280000, A317652, A317655, A317656, A317676.
%K nonn
%O 0,3
%A _Gus Wiseman_, Aug 17 2018
%E Terms a(8) and beyond from _Andrew Howroyd_, Dec 30 2020
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