A029868 Number of connected functions on n points with a loop of length 5.

1, 1, 4, 14, 46, 145, 440, 1315, 3877, 11315, 32792, 94529, 271510, 777764, 2223865, 6350657, 18120730, 51680249, 147359335, 420163711, 1198151432, 3417475326, 9750708533, 27831153091, 79471338455, 227032777454, 648896436944, 1855571389651, 5308837191604

Offset: 5

Author

Christian G. Bower

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 5..500
• C. G. Bower, Transforms

Crossrefs

Column 5 of A339428.

Programs

Mathematica

nn = 20; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol =
SolveAlways[
  0 == Series[
    f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
  x]; b = Flatten[Table[a[n], {n, 1, nn}] /. sol]; CoefficientList[
Series[CycleIndex[CyclicGroup[5], s] /.
   Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 5}], {x, 0,
nn}], x] (* Geoffrey Critzer, Aug 08 2013 *)

Formula

"CIK[ 5 ]" (necklace, indistinct, unlabeled, 5 parts) transform of A000081.

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

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020