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A029852 Number of connected functions on n points with a loop of length 3.

%I A029852 #26 Dec 25 2020 12:52:15
%S A029852 1,1,3,9,23,62,169,451,1217,3291,8916,24243,66155,181053,497134,
%T A029852 1369064,3780942,10469573,29063361,80867990,225508124,630145449,
%U A029852 1764240907,4948365051,13902893423,39124094362,110265280739,311208414556,879523722747,2488832434859
%N A029852 Number of connected functions on n points with a loop of length 3.
%H A029852 Alois P. Heinz, <a href="/A029852/b029852.txt">Table of n, a(n) for n = 3..1000</a>
%F A029852 G.f.: A(x) = ( B(x)^3 + 2*B(x^3) )/3 where B(x) is o.g.f. for A000081. - _Geoffrey Critzer_, Aug 09 2013
%F A029852 a(n) ~ A187770 * A051491^n / n^(3/2). - _Vaclav Kotesovec_, Dec 25 2020
%t A029852 nn = 20; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol =
%t A029852 SolveAlways[
%t A029852   0 == Series[
%t A029852     f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
%t A029852   x]; b = Flatten[Table[a[n], {n, 1, nn}] /. sol]; CoefficientList[
%t A029852 Series[CycleIndex[CyclicGroup[3], s] /.
%t A029852    Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 3}], {x, 0,
%t A029852 nn}], x] (* _Geoffrey Critzer_, Aug 08 2013 *)
%Y A029852 Column 3 of A339428.
%Y A029852 Cf. A000081.
%K A029852 nonn
%O A029852 3,3
%A A029852 _Christian G. Bower_