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

1, 1, 4, 11, 35, 97, 282, 792, 2243, 6275, 17602, 49206, 137713, 385208, 1078667, 3022342, 8478199, 23807190, 66932592, 188394855, 530911452, 1497892857, 4230987944, 11964356354, 33869704270, 95982410945, 272279600817, 773153124315, 2197492308752

Offset: 4

Author

Christian G. Bower

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 4..500

Crossrefs

Column 4 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[4], s] /.
   Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 4}], {x, 0,
nn}], x] (* Geoffrey Critzer, Aug 08 2013 *)

Formula

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

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