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A134959 Number of spanning hypertrees with n unlabeled vertices: analog of A035053 when edges of size 1 are allowed (with no two equal edges).

%I A134959 #17 Feb 10 2020 06:14:11
%S A134959 1,2,3,10,35,150,707,3700,20470,119260,719341,4466316,28367118,
%T A134959 183620874,1207563011,8049914664,54295152117,369981325578,
%U A134959 2544017965638,17633790542978,123108792874528,865045359778662,6114040341515978,43443726772579152,310195170229429300
%N A134959 Number of spanning hypertrees with n unlabeled vertices: analog of A035053 when edges of size 1 are allowed (with no two equal edges).
%H A134959 Andrew Howroyd, <a href="/A134959/b134959.txt">Table of n, a(n) for n = 0..200</a>
%F A134959 Inverse Euler transform of A134957. - _Gus Wiseman_, May 20 2018
%e A134959 From _Gus Wiseman_, May 20 2018: (Start)
%e A134959 Non-isomorphic representatives of the a(3) = 10 hypertrees are the following:
%e A134959   {{1,2,3}}
%e A134959   {{3},{1,2,3}}
%e A134959   {{1,3},{2,3}}
%e A134959   {{2},{3},{1,2,3}}
%e A134959   {{2},{1,3},{2,3}}
%e A134959   {{3},{1,3},{2,3}}
%e A134959   {{1},{2},{3},{1,2,3}}
%e A134959   {{1},{2},{1,3},{2,3}}
%e A134959   {{2},{3},{1,3},{2,3}}
%e A134959   {{1},{2},{3},{1,3},{2,3}}
%e A134959 (End)
%t A134959 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 A134959 EulerT[v_List] := With[{q = etr[v[[#]] &]}, q /@ Range[Length[v]]];
%t A134959 ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
%t A134959 b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];
%t A134959 seq[n_] := Module[{u = 2 b[n]}, 1 + x*ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&];
%t A134959 seq[25] (* _Jean-François Alcover_, Feb 10 2020, after _Andrew Howroyd_ *)
%o A134959 (PARI) \\ here b(n) is A318494 as vector
%o A134959 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A134959 b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
%o A134959 seq(n)={my(u=2*b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u)))} \\ _Andrew Howroyd_, Aug 27 2018
%Y A134959 Cf. A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304867, A304911, A318494.
%K A134959 nonn
%O A134959 0,2
%A A134959 _Don Knuth_, Jan 26 2008
%E A134959 Terms a(7) and beyond from _Andrew Howroyd_, Aug 27 2018