OFFSET
0,4
COMMENTS
Equivalently, the number of isomorphism class of transitive PSL_2(ZZ) actions on a finite set of size n.
Also the number of different connected trivalent diagrams of size n.
Also the number of (r,s) pair of permutations in S_n, up to simultaneous conjugation, which generate a transitive action and for which r is involutive i.e. r^2 = id and s is of weak order three i.e. s^3 = id.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
S. A. Vidal, Sur la Classification et le Dénombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison (in French), arXiv:0702223 [math.CO], 2006.
Qiaochu Yuan, Drawing subgroups of the modular group, Annoying Precision, Blog, November 29, 2015.
FORMULA
If A(z) = g.f. of a(n) and B(z) = g.f. of A121352 then A(z) = sum_{k > 0} mu(k)/k log(B(z^k)) (Moebius inversion formula).
MAPLE
with(numtheory, mobius) : mu := k -> `if`( k mod 2 = 0, 2/k, 1/k ) : nu := k -> `if`( k mod 3 = 0, 3/k, 1/k ) : u := (k, n) -> add(mu(k)^(n-2*k2)/(n-2*k2)!/k2!/(2*k)^k2, k2=0..floor(n/ 2)) ; v := (k, n) -> add(nu(k)^(n-3*k3)/(n-3*k3)!/k3!/(3*k)^k3, k3=0..floor(n/ 3)) ; N := 100 # For example. add(convert(taylor(log(add(n!*k^n*u(k, n)*v(k, n)*t^(k*n), n = 0..floor (N/k))), t=0, N+1), polynom), k=1..N) : lZF := sort (%, t, ascending) : add(mobius(k)/k*rem(subs(t=t^k, lZF), t^(N+1), t), k=1..N) : sort (%, t, ascending);
MATHEMATICA
max = 37; mu[k_] := If[Mod[k, 2] == 0, 2/k, 1/k]; nu[k_] := If[Mod[k, 3] == 0, 3/k, 1/k]; u[k_, n_] := Sum[ mu[k]^(n - 2*k2) / (((n - 2*k2)!*k2!)*(2*k)^k2), {k2, 0, Floor[n/2]}]; v[k_, n_] := Sum[ nu[k]^(n - 3*k3) / (((n - 3*k3)!*k3!)*(3*k)^k3), {k3, 0, Floor[n/3]}]; lZF[t_] = Sum[ Normal[ Series[ Log[ Sum[n!*k^n*u[k, n]*v[k, n]*t^(k*n), {n, 0, Floor[max/k]}]], {t, 0, max + 1}]], {k, 1, max}]; Rest[ CoefficientList[ Sum[ (MoebiusMu[k]*PolynomialMod[lZF[t^k], t^(max + 1)])/k, {k, 1, max}], t]] (* Jean-François Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Samuel A. Vidal, Jul 23 2006
EXTENSIONS
a(0)=1 prepended and a(38) onwards from Andrew Howroyd, Jan 29 2025
STATUS
approved