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%I #19 Oct 30 2023 13:25:15
%S 1,1,3,30,1912,18662590,12813206131799685,
%T 33758171486592987138461432668177794,
%U 1435913805026242504952006868879460423767388571975632398910903473535427583
%N Number of non-isomorphic connected set-systems covering n vertices.
%H Alois P. Heinz, <a href="/A323819/b323819.txt">Table of n, a(n) for n = 0..12</a>
%H Thomas Delacroix, <a href="https://hal-enpc.archives-ouvertes.fr/LAB-STICC_DMID/tel-03286641">Meaningful objective frequency-based interesting pattern mining</a>, Thesis, 2021.
%H Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, <a href="https://arxiv.org/abs/2310.15668">Hypergraph Motifs and Their Extensions Beyond Binary</a>, arXiv:2310.15668 [cs.SI], 2023.
%F Inverse Euler transform of A055621.
%e Non-isomorphic representatives of the a(3) = 30 set-systems:
%e {{1,2,3}}
%e {{3},{1,2,3}}
%e {{1,3},{2,3}}
%e {{2,3},{1,2,3}}
%e {{2},{3},{1,2,3}}
%e {{2},{1,3},{2,3}}
%e {{3},{1,3},{2,3}}
%e {{1},{2,3},{1,2,3}}
%e {{3},{2,3},{1,2,3}}
%e {{1,2},{1,3},{2,3}}
%e {{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2,3}}
%e {{1},{2},{1,3},{2,3}}
%e {{2},{3},{1,3},{2,3}}
%e {{1},{3},{2,3},{1,2,3}}
%e {{2},{3},{2,3},{1,2,3}}
%e {{3},{1,2},{1,3},{2,3}}
%e {{2},{1,3},{2,3},{1,2,3}}
%e {{3},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,3},{2,3}}
%e {{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{2,3},{1,2,3}}
%e {{2},{3},{1,2},{1,3},{2,3}}
%e {{1},{2},{1,3},{2,3},{1,2,3}}
%e {{2},{3},{1,3},{2,3},{1,2,3}}
%e {{3},{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3},{2,3}}
%e {{1},{2},{3},{1,3},{2,3},{1,2,3}}
%e {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%t nmax = 12;
%t 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}]]];
%t f[n_] := If[n == 0, 2, b[n, n, {}] - b[n - 1, n - 1, {}]]/2;
%t A055621 = f /@ Range[0, nmax];
%t mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t 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]];
%t Join[{1}, EULERi[A055621 // Rest]] (* _Jean-François Alcover_, Jan 31 2020, after _Alois P. Heinz_ in A055621 *)
%Y Cf. A000295, A003465, A016031, A048143, A055621 (not necessarily connected), A293510, A317795, A323817, A323818 (labeled case).
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 30 2019
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