%I #26 Dec 25 2020 12:52:15

%S 1,1,3,9,23,62,169,451,1217,3291,8916,24243,66155,181053,497134,

%T 1369064,3780942,10469573,29063361,80867990,225508124,630145449,

%U 1764240907,4948365051,13902893423,39124094362,110265280739,311208414556,879523722747,2488832434859

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

%H Alois P. Heinz, <a href="/A029852/b029852.txt">Table of n, a(n) for n = 3..1000</a>

%F 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 a(n) ~ A187770 * A051491^n / n^(3/2). - _Vaclav Kotesovec_, Dec 25 2020

%t nn = 20; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol =

%t SolveAlways[

%t 0 == Series[

%t f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],

%t x]; b = Flatten[Table[a[n], {n, 1, nn}] /. sol]; CoefficientList[

%t Series[CycleIndex[CyclicGroup[3], s] /.

%t Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 3}], {x, 0,

%t nn}], x] (* _Geoffrey Critzer_, Aug 08 2013 *)

%Y Column 3 of A339428.

%Y Cf. A000081.

%K nonn

%O 3,3

%A _Christian G. Bower_