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A304912 Number of non-isomorphic spanning hyperforests of weight n.

%I A304912 #21 Feb 10 2020 02:22:55
%S A304912 1,1,2,3,6,9,18,29,56,97,186,337,657,1238,2442,4768,9569,19174,39151,
%T A304912 80154,166211,346239,727853,1537611,3270710,6989669,15018389,32405378,
%U A304912 70230238,152772075,333552711,730632928,1605459844,3537861659,7817447580,17317397837
%N A304912 Number of non-isomorphic spanning hyperforests of weight n.
%C A304912 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).
%H A304912 Andrew Howroyd, <a href="/A304912/b304912.txt">Table of n, a(n) for n = 0..500</a>
%F A304912 Euler transform of A304867.
%e A304912 The a(6) = 18 spanning hyperforests are the following:
%e A304912   {{1,2,3,4,5,6}}
%e A304912   {{1},{2,3,4,5,6}}
%e A304912   {{1,2},{3,4,5,6}}
%e A304912   {{1,5},{2,3,4,5}}
%e A304912   {{1,2,3},{4,5,6}}
%e A304912   {{1,2,5},{3,4,5}}
%e A304912   {{1},{2},{3,4,5,6}}
%e A304912   {{1},{2,3},{4,5,6}}
%e A304912   {{1},{2,5},{3,4,5}}
%e A304912   {{1,2},{3,4},{5,6}}
%e A304912   {{1,2},{3,5},{4,5}}
%e A304912   {{1,3},{2,4},{3,4}}
%e A304912   {{1,4},{2,4},{3,4}}
%e A304912   {{1},{2},{3},{4,5,6}}
%e A304912   {{1},{2},{3,4},{5,6}}
%e A304912   {{1},{2},{3,5},{4,5}}
%e A304912   {{1},{2},{3},{4},{5,6}}
%e A304912   {{1},{2},{3},{4},{5},{6}}
%t A304912 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];
%t A304912 EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
%t A304912 ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
%t A304912 c[n_] := Module[{v = {1}}, For[i = 1, i <= Ceiling[n/2], i++, v = Join[{1}, EulerT[Join[{0}, EulerT[v]]]]]; v];
%t A304912 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]&];
%t A304912 seq[36] (* _Jean-François Alcover_, Feb 09 2020, after _Andrew Howroyd_ *)
%o A304912 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A304912 c(n)={my(v=[1]); for(i=2, ceil(n/2), v=concat([1], EulerT(concat([0], EulerT(v))))); v}
%o A304912 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
%Y A304912 Cf. A007716, A035053, A048143, A054921, A134954, A134955, A134957, A144959, A286520, A293993, A293994, A304911, A304867.
%K A304912 nonn
%O A304912 0,3
%A A304912 _Gus Wiseman_, May 20 2018
%E A304912 Terms a(10) and beyond from _Andrew Howroyd_, Aug 29 2018