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A134957 Number of hyperforests with n unlabeled vertices: analog of A134955 when edges of size 1 are allowed (with no two equal edges).

%I A134957 #15 Feb 10 2020 06:23:16
%S A134957 1,2,6,20,75,310,1422,7094,37877,213610,1256422,7641700,47735075,
%T A134957 304766742,1981348605,13079643892,87480944764,591771554768,
%U A134957 4042991170169,27864757592632,193549452132550,1353816898675732,9529263306483357,67457934248821368,480019516988969011
%N A134957 Number of hyperforests with n unlabeled vertices: analog of A134955 when edges of size 1 are allowed (with no two equal edges).
%H A134957 Andrew Howroyd, <a href="/A134957/b134957.txt">Table of n, a(n) for n = 0..200</a>
%F A134957 Euler transform of A134959. - _Gus Wiseman_, May 20 2018
%e A134957 From _Gus Wiseman_, May 20 2018: (Start)
%e A134957 Non-isomorphic representatives of the a(3) = 20 hyperforests are the following:
%e A134957   {}
%e A134957   {{1}}
%e A134957   {{1,2}}
%e A134957   {{1,2,3}}
%e A134957   {{1},{2}}
%e A134957   {{1},{2,3}}
%e A134957   {{2},{1,2}}
%e A134957   {{3},{1,2,3}}
%e A134957   {{1,3},{2,3}}
%e A134957   {{1},{2},{3}}
%e A134957   {{1},{2},{1,2}}
%e A134957   {{1},{3},{2,3}}
%e A134957   {{2},{3},{1,2,3}}
%e A134957   {{2},{1,3},{2,3}}
%e A134957   {{3},{1,3},{2,3}}
%e A134957   {{1,2},{1,3},{2,3}}
%e A134957   {{1},{2},{3},{2,3}}
%e A134957   {{1},{2},{3},{1,2,3}}
%e A134957   {{1},{2},{1,3},{2,3}}
%e A134957   {{2},{3},{1,3},{2,3}}
%e A134957   {{3},{1,2},{1,3},{2,3}}
%e A134957   {{1},{2},{3},{1,3},{2,3}}
%e A134957   {{2},{3},{1,2},{1,3},{2,3}}
%e A134957   {{1},{2},{3},{1,2},{1,3},{2,3}}
%e A134957 (End)
%t A134957 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 A134957 EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
%t A134957 ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
%t A134957 b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];
%t A134957 seq[n_] := Module[{u = 2 b[n]}, Join[{1}, EulerT[ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&]]];
%t A134957 seq[24] (* _Jean-François Alcover_, Feb 10 2020, after _Andrew Howroyd_ *)
%o A134957 (PARI) \\ here b(n) is A318494 as vector
%o A134957 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A134957 b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
%o A134957 seq(n)={my(u=2*b(n)); concat([1], EulerT(Vec(Ser(EulerT(u))*(1-x*Ser(u)))))} \\ _Andrew Howroyd_, Aug 27 2018
%Y A134957 Cf. A030019, A035053, A048143, A054921, A134955, A134957, A134959, A144959, A304867, A304911.
%K A134957 nonn
%O A134957 0,2
%A A134957 _Don Knuth_, Jan 26 2008
%E A134957 Terms a(7) and beyond from _Andrew Howroyd_, Aug 27 2018