A318869 Inverse Euler transform of A122082.

1, 2, 2, 8, 37, 270, 3049, 56576, 1795917, 100752972, 10189362127, 1879720761478, 637617233746767, 400169631649617320, 467115844246535037894, 1018822456144129013291710, 4169121243929999971120036590, 32126195519194538602120203293590

Offset: 0

Author

Andrew Howroyd, Sep 04 2018

Keywords

• nonn

Comments

This sequence is an intermediate step in the computation of A005142 and A123549.

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

Crossrefs

Cf. A005142, A122082, A123549, A318870.

Programs

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 *)