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A029853 Number of connected functions on n points with a loop of length 4.

%I A029853 #17 Dec 25 2020 12:56:34
%S A029853 1,1,4,11,35,97,282,792,2243,6275,17602,49206,137713,385208,1078667,
%T A029853 3022342,8478199,23807190,66932592,188394855,530911452,1497892857,
%U A029853 4230987944,11964356354,33869704270,95982410945,272279600817,773153124315,2197492308752
%N A029853 Number of connected functions on n points with a loop of length 4.
%H A029853 Andrew Howroyd, <a href="/A029853/b029853.txt">Table of n, a(n) for n = 4..500</a>
%F A029853 G.f.: A(x) = ( B(x)^4 + B(x^2)^2 + 2*B(x^4) )/4 where B(x) is the o.g.f. for A000081. - _Geoffrey Critzer_, Aug 09 2013
%F A029853 a(n) ~ A187770 * A051491^n / n^(3/2). - _Vaclav Kotesovec_, Dec 25 2020
%t A029853 nn = 20; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol =
%t A029853 SolveAlways[
%t A029853   0 == Series[
%t A029853     f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
%t A029853   x]; b = Flatten[Table[a[n], {n, 1, nn}] /. sol]; CoefficientList[
%t A029853 Series[CycleIndex[CyclicGroup[4], s] /.
%t A029853    Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 4}], {x, 0,
%t A029853 nn}], x] (* _Geoffrey Critzer_, Aug 08 2013 *)
%Y A029853 Column 4 of A339428.
%K A029853 nonn
%O A029853 4,3
%A A029853 _Christian G. Bower_