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A317881
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Number of series-reduced free pure identity multifunctions (with empty expressions allowed) with one atom and n positions.
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8
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1, 1, 1, 1, 3, 7, 15, 37, 91, 231, 593, 1557, 4111, 10941, 29295, 79087, 215015, 587463, 1611985, 4441473, 12284513, 34095797, 94931525, 265061363, 742029431, 2082310665, 5856540305, 16505796865, 46608877763, 131850193107, 373612733107, 1060339387939, 3013758348317
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OFFSET
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1,5
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COMMENTS
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A series-reduced series-reduced free pure identity multifunction (with empty expressions allowed) (SRIM) 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 SRIM, k is an integer greater than or equal to 0 but not equal to 1, each of the g_i for i = 1, ..., k >= 0 is an SRIM, and for i != j we have g_i != g_j. The number of positions in an SRIM is the number of brackets [...] plus the number of o's.
Also the number of series-reduced identity Mathematica expressions with one atom and n positions.
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LINKS
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EXAMPLE
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The a(6) = 7 SRIMs:
o[o[][],o]
o[o,o[][]]
o[][o[],o]
o[][o,o[]]
o[o[],o][]
o[o,o[]][]
o[][][][][]
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MATHEMATICA
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allIdExprSR[n_]:=If[n==1, {"o"}, Join@@Cases[Table[PR[k, n-k-1], {k, n-1}], PR[h_, g_]:>Join@@Table[Apply@@@Tuples[{allIdExprSR[h], Select[Tuples[allIdExprSR/@p], UnsameQ@@#&]}], {p, If[g==0, {{}}, Join@@Permutations/@Rest[IntegerPartitions[g]]]}]]];
Table[Length[allIdExprSR[n]], {n, 12}]
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PROG
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(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[k]))); v} \\ Andrew Howroyd, Sep 01 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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