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%I #18 Dec 25 2020 12:58:37
%S 1,1,4,14,46,145,440,1315,3877,11315,32792,94529,271510,777764,
%T 2223865,6350657,18120730,51680249,147359335,420163711,1198151432,
%U 3417475326,9750708533,27831153091,79471338455,227032777454,648896436944,1855571389651,5308837191604
%N Number of connected functions on n points with a loop of length 5.
%H Andrew Howroyd, <a href="/A029868/b029868.txt">Table of n, a(n) for n = 5..500</a>
%H C. G. Bower, <a href="/transforms2.html">Transforms</a>
%F "CIK[ 5 ]" (necklace, indistinct, unlabeled, 5 parts) transform of A000081.
%F G.f.: A(x) = ( B(x)^5 +4*B(x^5) )/5 where B(x) is the 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[5], s] /.
%t Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 5}], {x, 0,
%t nn}], x] (* _Geoffrey Critzer_, Aug 08 2013 *)
%Y Column 5 of A339428.
%K nonn
%O 5,3
%A _Christian G. Bower_
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