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A317876 Number of free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions. 8
1, 1, 2, 4, 10, 25, 67, 184, 519, 1489, 4342, 12812, 38207, 114934, 348397, 1063050, 3262588, 10064645, 31190985, 97061431, 303165207, 950115502, 2986817742, 9415920424, 29760442192, 94286758293, 299377379027, 952521579944, 3036380284111, 9696325863803 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A free pure symmetric identity multifunction (with empty expressions allowed) (FOI) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where h is an FOI, each of the g_i for i = 1, ..., k >= 0 is an FOI, 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 an FOI is the number of brackets [...] plus the number of o's.

Also the number of free orderless identity Mathematica expressions with one atom and n positions.

LINKS

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

FORMULA

From Ilya Gutkovskiy, Apr 30 2019: (Start)

G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} (-1)^(k+1)*A(x^k)/k)).

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + (Sum_{n>=1} a(n)*x^n) * Product_{n>=1} (1 + x^n)^a(n)). (End)

EXAMPLE

The a(5) = 10 FOIs:

  o[o[o]]

  o[o][o]

  o[o[][]]

  o[o,o[]]

  o[][o[]]

  o[][][o]

  o[o[]][]

  o[][o][]

  o[o][][]

  o[][][][]

MATHEMATICA

allIdExpr[n_]:=If[n==1, {"o"}, Join@@Cases[Table[PR[k, n-k-1], {k, n-1}], PR[h_, g_]:>Join@@Table[Apply@@@Tuples[{allIdExpr[h], Select[Union[Sort/@Tuples[allIdExpr/@p]], UnsameQ@@#&]}], {p, IntegerPartitions[g]}]]];

Table[Length[allIdExpr[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, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

CROSSREFS

Cf. A000081, A004111, A052893, A053492, A277996, A280000, A317652, A317653, A317654, A317875.

Cf. A317877, A317878, A317879, A317880, A317881.

Sequence in context: A195981 A124500 A220872 * A124501 A124344 A049125

Adjacent sequences:  A317873 A317874 A317875 * A317877 A317878 A317879

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 09 2018

EXTENSIONS

Terms a(16) and beyond from Andrew Howroyd, Aug 19 2018

STATUS

approved

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Last modified June 25 12:55 EDT 2019. Contains 324352 sequences. (Running on oeis4.)