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Inverse Euler transform of A122082.
4

%I #13 Sep 13 2018 02:42:23

%S 1,2,2,8,37,270,3049,56576,1795917,100752972,10189362127,

%T 1879720761478,637617233746767,400169631649617320,

%U 467115844246535037894,1018822456144129013291710,4169121243929999971120036590,32126195519194538602120203293590

%N Inverse Euler transform of A122082.

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

%C 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 Andrew Howroyd, <a href="/A318869/b318869.txt">Table of n, a(n) for n = 0..50</a>

%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 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 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i-1}] + Total @ Quotient[v+1, 2]

%t b[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);

%t Join[{1}, EULERi[Array[b, 20]]] (* _Jean-François Alcover_, Sep 13 2018, after _Andrew Howroyd_ *)

%Y Cf. A005142, A122082, A123549, A318870.

%K nonn

%O 0,2

%A _Andrew Howroyd_, Sep 04 2018