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 A318869 Inverse Euler transform of A122082. 4
 1, 2, 2, 8, 37, 270, 3049, 56576, 1795917, 100752972, 10189362127, 1879720761478, 637617233746767, 400169631649617320, 467115844246535037894, 1018822456144129013291710, 4169121243929999971120036590, 32126195519194538602120203293590 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 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 Cf. A005142, A122082, A123549, A318870. Sequence in context: A219348 A009543 A102647 * A060224 A232980 A212307 Adjacent sequences:  A318866 A318867 A318868 * A318870 A318871 A318872 KEYWORD nonn AUTHOR Andrew Howroyd, Sep 04 2018 STATUS approved

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Last modified June 24 03:27 EDT 2021. Contains 345415 sequences. (Running on oeis4.)