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%I #29 Oct 29 2018 07:16:14
%S 1,1,8,48,120,2640,30240,201600,4838400,96163200,1037836800,
%T 30496435200,828193766400,13686991718400,450537408921600,
%U 15880397524992000,356398802952192000,13410127414075392000,569542360114151424000,16614774394239909888000
%N Number of transitive PSL_2(ZZ) actions on a finite labeled set of size n.
%C Equivalently, the number of different connected labeled trivalent diagrams of size n.
%C Also the number of (r,s) pair of permutations in S_n, 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 Denombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison</a>, (in French), arXiv:math/0702223 [math.CO] 2006.
%F If A(z) = g.f. of a(n) and B(z) = g.f. of A121357 then A(z) = log(B(z)).
%p N := 100 : exs2:=sort(convert(taylor(exp(t+t^2/2),t,N+1),polynom),t, ascending) : exs3:=sort(convert(taylor(exp(t+t^3/3),t,N+1),polynom),t, ascending) : exs23:=sort(add(op(n+1,exs2)*op(n+1,exs3)/(t^n/ n!),n=0..N),t, ascending) : logexs23:=sort(convert(taylor(log(exs23),t,N+1),polynom),t, ascending) : sort(add(op(n,logexs23)*n!,n=1..N),t, ascending);
%p # Alternatively:
%p A121355_list := proc(len) local s,p; s := f -> seq(n!*coeff(series(f,z,n+1),z,n), n=0..len); p := m -> s(exp(z+z^m/m)); s(log(add((p(2)[n+1]*p(3)[n+1])*z^n/n!, n=0..len))) end: # _Peter Luschny_, Nov 16 2015
%t m = 19; exs2 = Series[Exp[t + t^2/2], {t, 0, m + 1}] // Normal; exs3 = Series[Exp[t + t^3/3], {t, 0, m + 1}] // Normal; exs23 = Sum[exs2[[n + 1]]*exs3[[n + 1]]/(t^n/n!), {n, 0, m}]; logexs23 = Series[Log[exs23], {t, 0, m}] // Normal; CoefficientList[logexs23, t]*Range[0, m]! // Rest (* _Jean-François Alcover_, Sep 06 2013, translated and adapted from Samuel Vidal's Maple program *)
%o (PARI) N=20; x='x+O('x^(N+1));
%o A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));
%o Vec(serlaplace(log(serconvol(A121357_ser, exp(x))))) \\ _Gheorghe Coserea_, May 10 2017
%Y Connected version of A121357.
%Y Labeled version of A121350.
%Y Cf. also A005133, A121352, A121356.
%K nonn
%O 1,3
%A _Samuel A. Vidal_, Jul 23 2006
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