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A317878 Number of free pure symmetric identity multifunctions with one atom and n positions. 8
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A129157 A086905 A167638 * A209108 A269019 A184234
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018
STATUS
approved

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Last modified July 23 23:47 EDT 2024. Contains 374575 sequences. (Running on oeis4.)