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G.f.: x * (d/dx) x * Product_{k>=1} (1 + x^k)^(a(k)/k).
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%I #8 May 27 2019 18:23:32

%S 1,2,3,8,15,36,84,200,468,1130,2717,6576,15938,38780,94485,230816,

%T 564553,1383318,3393742,8336960,20502216,50472928,124369832,306729456,

%U 757078000,1870040822,4622317812,11432698704,28294211920,70063292310,173584768088,430276174016,1067049650238

%N G.f.: x * (d/dx) x * Product_{k>=1} (1 + x^k)^(a(k)/k).

%F L.g.f.: x * exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*a(d) ) * x^k/k).

%F a(n) = n * A004111(n).

%t a[n_] := a[n] = SeriesCoefficient[x D[x Product[(1 + x^k)^(a[k]/k), {k, 1, n - 1}], x], {x, 0, n}]; Table[a[n], {n, 1, 33}]

%t a[n_] := a[n] = n SeriesCoefficient[x Exp[Sum[Sum[(-1)^(k/d + 1) a[d], {d, Divisors[k]}] x^k/k, {k, 1, n - 1}]], {x, 0, n}]; Table[a[n], {n, 1, 33}]

%t a[n_] := a[n] = Sum[a[n - k] Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}], {k, 1, n - 1}]/(n - 1); a[1] = 1; Table[n a[n], {n, 1, 33}]

%Y Cf. A004111, A055544.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, May 26 2019