proposed

approved

editing

proposed

1, 0, 1, 0, 2, 1, 5, 5, 15, 23, 54, 98, 212, 420, 886, 1822, 3838, 8046, 17029, 36097, 76889, 164245, 351971, 756341, 1629389, 3518643, 7614717, 16512962, 35875986, 78082171, 170219300, 371651968, 812624721, 1779240627, 3900634491, 8561723769, 18814112811

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

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

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

nonn,more,new

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018

approved

editing

proposed

approved

editing

proposed

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

allocated for Gus WisemanNumber of free pure symmetric identity multifunctions with one atom and n positions.

1, 0, 1, 0, 2, 1, 5, 5, 15, 23, 54, 98

1,5

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.

The a(8) = 5 SIMs:

o[o[o,o[o]]]

o[o,o[o[o]]]

o[o,o[o][o]]

o[o][o,o[o]]

o[o,o[o]][o]

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]}]]];

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

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

allocated

nonn,more

Gus Wiseman, Aug 09 2018

approved

editing

allocated for Gus Wiseman

allocated

approved