OFFSET
0,2
COMMENTS
The combinatoric interpretation is that of connected bicolored graphs on 2n nodes which are invariant when the two color classes are interchanged plus pairs of identical connected bicolored graphs on n nodes each which are not invariant when the two color classes are interchanged. The former is A123549(n) and the later is A005142(n) for odd n and A005142(n) - A123549(n/2) for even n.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
MATHEMATICA
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]];
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i-1}] + Total @ Quotient[v+1, 2]
b[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Join[{1}, EULERi[Array[b, 20]]] (* Jean-François Alcover, Sep 13 2018, after Andrew Howroyd *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Sep 04 2018
STATUS
approved