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A318869 Inverse Euler transform of A122082.

%I A318869 #13 Sep 13 2018 02:42:23
%S A318869 1,2,2,8,37,270,3049,56576,1795917,100752972,10189362127,
%T A318869 1879720761478,637617233746767,400169631649617320,
%U A318869 467115844246535037894,1018822456144129013291710,4169121243929999971120036590,32126195519194538602120203293590
%N A318869 Inverse Euler transform of A122082.
%C A318869 This sequence is an intermediate step in the computation of A005142 and A123549.
%C A318869 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.
%H A318869 Andrew Howroyd, <a href="/A318869/b318869.txt">Table of n, a(n) for n = 0..50</a>
%t A318869 mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t A318869 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 A318869 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];
%t A318869 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i-1}] + Total @ Quotient[v+1, 2]
%t A318869 b[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
%t A318869 Join[{1}, EULERi[Array[b, 20]]] (* _Jean-François Alcover_, Sep 13 2018, after _Andrew Howroyd_ *)
%Y A318869 Cf. A005142, A122082, A123549, A318870.
%K A318869 nonn
%O A318869 0,2
%A A318869 _Andrew Howroyd_, Sep 04 2018