A317878 Number of free pure symmetric identity multifunctions with one atom and n positions.

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

Offset

1,5

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Example

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]

Mathematica

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}]

Prog

(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

Crossrefs

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

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

Sequence in context: A129157 A086905 A167638 * A209108 A269019 A184234

Adjacent sequences: A317875 A317876 A317877 * A317879 A317880 A317881

Keyword

nonn

Author

Gus Wiseman, Aug 09 2018

Extensions

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

Status

approved