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A323819 Number of non-isomorphic connected set-systems covering n vertices. 6
1, 1, 3, 30, 1912, 18662590, 12813206131799685, 33758171486592987138461432668177794, 1435913805026242504952006868879460423767388571975632398910903473535427583 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..12

FORMULA

Inverse Euler transform of A055621.

EXAMPLE

Non-isomorphic representatives of the a(3) = 30 set-systems:

  {{1,2,3}}

  {{3},{1,2,3}}

  {{1,3},{2,3}}

  {{2,3},{1,2,3}}

  {{2},{3},{1,2,3}}

  {{2},{1,3},{2,3}}

  {{3},{1,3},{2,3}}

  {{1},{2,3},{1,2,3}}

  {{3},{2,3},{1,2,3}}

  {{1,2},{1,3},{2,3}}

  {{1,3},{2,3},{1,2,3}}

  {{1},{2},{3},{1,2,3}}

  {{1},{2},{1,3},{2,3}}

  {{2},{3},{1,3},{2,3}}

  {{1},{3},{2,3},{1,2,3}}

  {{2},{3},{2,3},{1,2,3}}

  {{3},{1,2},{1,3},{2,3}}

  {{2},{1,3},{2,3},{1,2,3}}

  {{3},{1,3},{2,3},{1,2,3}}

  {{1},{2},{3},{1,3},{2,3}}

  {{1,2},{1,3},{2,3},{1,2,3}}

  {{1},{2},{3},{2,3},{1,2,3}}

  {{2},{3},{1,2},{1,3},{2,3}}

  {{1},{2},{1,3},{2,3},{1,2,3}}

  {{2},{3},{1,3},{2,3},{1,2,3}}

  {{3},{1,2},{1,3},{2,3},{1,2,3}}

  {{1},{2},{3},{1,2},{1,3},{2,3}}

  {{1},{2},{3},{1,3},{2,3},{1,2,3}}

  {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

MATHEMATICA

nmax = 12;

b[n_, i_, l_] := b[n, i, l] = If[n == 0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]], If[i < 1, 0, Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];

f[n_] := If[n == 0, 2, b[n, n, {}] - b[n - 1, n - 1, {}]]/2;

A055621 = f /@ Range[0, nmax];

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];

EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];

Join[{1}, EULERi[A055621 // Rest]] (* Jean-Fran├žois Alcover, Jan 31 2020, after Alois P. Heinz in A055621 *)

CROSSREFS

Cf. A000295, A003465, A016031, A048143, A055621 (not necessarily connected), A293510, A317795, A323817, A323818 (labeled case).

Sequence in context: A203415 A203520 A238767 * A308228 A208789 A296173

Adjacent sequences:  A323816 A323817 A323818 * A323820 A323821 A323822

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 30 2019

STATUS

approved

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Last modified August 4 15:42 EDT 2020. Contains 336201 sequences. (Running on oeis4.)