OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..12
Thomas Delacroix, Meaningful objective frequency-based interesting pattern mining, Thesis, 2021.
Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, Hypergraph Motifs and Their Extensions Beyond Binary, arXiv:2310.15668 [cs.SI], 2023.
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-François Alcover, Jan 31 2020, after Alois P. Heinz in A055621 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 30 2019
STATUS
approved