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%I #27 Oct 27 2018 09:43:38
%S 1,1,2,2,1,8,6,7,14,27,26,80,133,170,348,765,1002,2176,4682,6931,
%T 13740,31085,48652,96682,217152,362779,707590,1597130,2789797,5449439,
%U 12233848,22245655,43480188,97330468,182619250,358968639,800299302
%N Number of conjugacy class of index n subgroups in PSL_2 (ZZ).
%C Equivalently, the number of isomorphism class of transitive PSL_2(ZZ) actions on a finite set of size n.
%C Also the number of different connected trivalent diagrams of size n.
%C 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.
%H S. A. Vidal, <a href="https://arxiv.org/abs/math/0702223">Sur la Classification et le Dénombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison</a> (in French), arXiv:0702223 [math.CO], 2006.
%H Qiaochu Yuan, <a href="https://qchu.wordpress.com/2015/11/29/drawing-subgroups-of-the-modular-group/">Drawing subgroups of the modular group</a>, Annoying Precision, Blog, November 29, 2015.
%F 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)
%p 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);
%t 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 *)
%Y Connected version of A121352.
%Y Unlabeled version of A121355.
%Y Cf. also A005133, A121356, A121357.
%K nonn
%O 1,3
%A _Samuel A. Vidal_, Jul 23 2006
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