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A304912
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Number of non-isomorphic spanning hyperforests of weight n.
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18
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1, 1, 2, 3, 6, 9, 18, 29, 56, 97, 186, 337, 657, 1238, 2442, 4768, 9569, 19174, 39151, 80154, 166211, 346239, 727853, 1537611, 3270710, 6989669, 15018389, 32405378, 70230238, 152772075, 333552711, 730632928, 1605459844, 3537861659, 7817447580, 17317397837
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OFFSET
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0,3
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COMMENTS
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A spanning hyperforest is an antichain of finite nonempty sets, which cover a set of n vertices, whose connected components are hypertrees (see A304867). The weight of a hypertree is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices (see A134957).
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LINKS
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FORMULA
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EXAMPLE
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The a(6) = 18 spanning hyperforests are the following:
{{1,2,3,4,5,6}}
{{1},{2,3,4,5,6}}
{{1,2},{3,4,5,6}}
{{1,5},{2,3,4,5}}
{{1,2,3},{4,5,6}}
{{1,2,5},{3,4,5}}
{{1},{2},{3,4,5,6}}
{{1},{2,3},{4,5,6}}
{{1},{2,5},{3,4,5}}
{{1,2},{3,4},{5,6}}
{{1,2},{3,5},{4,5}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{1},{2},{3},{4,5,6}}
{{1},{2},{3,4},{5,6}}
{{1},{2},{3,5},{4,5}}
{{1},{2},{3},{4},{5,6}}
{{1},{2},{3},{4},{5},{6}}
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MATHEMATICA
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etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
c[n_] := Module[{v = {1}}, For[i = 1, i <= Ceiling[n/2], i++, v = Join[{1}, EulerT[Join[{0}, EulerT[v]]]]]; v];
seq[n_] := Module[{u = c[n]}, x*ser[EulerT[u]]*(1 - x*ser[u]) + (1 - x)* ser[u] + x + O[x]^n // CoefficientList[#, x]& // Rest // EulerT // Prepend[#, 1]&];
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PROG
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(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
c(n)={my(v=[1]); for(i=2, ceil(n/2), v=concat([1], EulerT(concat([0], EulerT(v))))); v}
seq(n)={my(u=c(n)); concat([1], EulerT(Vec(x*Ser(EulerT(u))*(1-x*Ser(u)) + (1 - x)*(Ser(u) - 1)+ O(x*x^n))))} \\ Andrew Howroyd, Aug 29 2018
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CROSSREFS
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Cf. A007716, A035053, A048143, A054921, A134954, A134955, A134957, A144959, A286520, A293993, A293994, A304911, A304867.
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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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