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A317884
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Number of series-reduced achiral free pure multifunctions (with empty expressions allowed) with one atom and n positions.
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6
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1, 1, 1, 2, 4, 8, 14, 26, 47, 87, 160, 295, 540, 997, 1832, 3369, 6197, 11406, 20975, 38594, 70991, 130610, 240275, 442043, 813184, 1496053, 2752251, 5063319, 9314879, 17136632, 31526032, 57998423, 106699160, 196294065, 361120800, 664352454, 1222204958
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OFFSET
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1,4
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COMMENTS
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A series-reduced achiral expression (SRAE) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty but not unitary expression of the form h[g, ..., g], where h and g are SRAEs. The number of positions in an SRAE is the number of brackets [...] plus the number of o's.
Also the number of series-reduced achiral Mathematica expressions with one atom and n positions.
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LINKS
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FORMULA
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a(1) = 1; a(n > 1) = a(n-1) + Sum_{0 < k < n-1} a(k) * Sum_{d|(n-k-1), d < n-k-1} a(d).
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EXAMPLE
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The a(6) = 8 SRAEs:
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][][]
o[][][][][]
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MAPLE
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a:= proc(n) option remember; `if`(n=1, 1, a(n-1)+add(a(j)*add(
a(d), d=numtheory[divisors](n-j-1) minus {n-j-1}), j=1..n-1))
end:
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MATHEMATICA
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allAchExprSR[n_] := If[n == 1, {"o"}, Join @@ Cases[Table[PR[k, n - k - 1], {k, n - 1}], PR[h_, g_] :> Join @@ Table[Apply @@@ Tuples[{allAchExprSR[h], Select[Tuples[allAchExprSR /@ p], SameQ @@ # &]}], {p, If[g == 0, {{}}, Join @@ Permutations /@ Rest[IntegerPartitions[g]]]}]]]; Table[Length[allAchExprSR[n]], {n, 12}]
(* Second program: *)
a[n_] := a[n] = If[n == 1, 1, a[n-1] + Sum[a[j]*DivisorSum[
n-j-1, If[# < n-j-1, a[#], 0]&], {j, 1, n-2}]];
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PROG
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(PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*(1 + sum(k=2, n-2, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018
(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=v[n-1] + sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, if(d<n-i-1, v[d], 0)))); v} \\ Andrew Howroyd, Aug 19 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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