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

1, 2, 3, 10, 35, 150, 707, 3700, 20470, 119260, 719341, 4466316, 28367118, 183620874, 1207563011, 8049914664, 54295152117, 369981325578, 2544017965638, 17633790542978, 123108792874528, 865045359778662, 6114040341515978, 43443726772579152, 310195170229429300

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

Inverse Euler transform of A134957. - Gus Wiseman, May 20 2018

Example

From Gus Wiseman, May 20 2018: (Start)
Non-isomorphic representatives of the a(3) = 10 hypertrees are the following:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
(End)

Mathematica

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];
b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];
seq[n_] := Module[{u = 2 b[n]}, 1 + x*ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&];
seq[25] (* Jean-François Alcover, Feb 10 2020, after Andrew Howroyd *)

Prog

(PARI) \\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u)))} \\ Andrew Howroyd, Aug 27 2018

Crossrefs

Cf. A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304867, A304911, A318494.

Sequence in context: A059735 A358213 A356926 * A270367 A056607 A278051

Adjacent sequences: A134956 A134957 A134958 * A134960 A134961 A134962

Keyword

nonn

Author

Don Knuth, Jan 26 2008

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

Status

approved