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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
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
Sequence in context: A195981 A124500 A220872 * A124501 A124344 A049125
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 12 23:30 EDT 2024. Contains 373362 sequences. (Running on oeis4.)