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A317879 Number of free pure identity multifunctions (with empty expressions allowed) with one atom and n positions. 8
1, 1, 2, 4, 11, 29, 83, 251, 767, 2403, 7652, 24758, 80875, 266803, 887330, 2972108, 10016981, 33942461, 115572864, 395226810, 1356840007, 4674552089, 16156355357, 56003840659, 194651585875, 678220460687, 2368505647624, 8288873657180, 29064904732911 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A free pure identity multifunction (with empty expressions allowed) (IME) 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 IME, each of the g_i for i = 1, ..., k >= 0 is an IME, and for i != j we have g_i != g_j. The number of positions in an IME is the number of brackets [...] plus the number of o's.
Also the number of identity Mathematica expressions with one atom and n positions.
LINKS
EXAMPLE
The a(5) = 11 IMEs:
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
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[Tuples[allIdExpr/@p], UnsameQ@@#&]}], {p, Join@@Permutations/@IntegerPartitions[g]}]]];
Table[Length[allIdExpr[n]], {n, 12}]
PROG
(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, my(p=prod(k=1, n, 1 + sum(i=1, n\k, binomial(v[k], i)*x^(i*k)*y^i) + O(x*x^n))); v[n]=v[n-1]+sum(k=1, n-2, v[n-k-1]*subst(serlaplace(y^0*polcoef(p, k)), y, 1))); v} \\ Andrew Howroyd, Sep 01 2018
CROSSREFS
Sequence in context: A328139 A369359 A148141 * A148142 A148143 A148144
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Sep 01 2018
STATUS
approved

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