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Number of transitive PSL_2(ZZ) actions on a finite dotted and labeled set of size n.
5

%I #23 Oct 29 2018 07:16:20

%S 1,2,24,192,600,15840,211680,1612800,43545600,961632000,11416204800,

%T 365957222400,10766518963200,191617884057600,6758061133824000,

%U 254086360399872000,6058779650187264000,241382293453357056000

%N Number of transitive PSL_2(ZZ) actions on a finite dotted and labeled set of size n.

%C "Dotted" means having a distinguished element. - _N. J. A. Sloane_, Feb 06 2012

%C Equivalently, the number of different connected, dotted and labeled trivalent diagrams of size n.

%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:math/0702223 [math.CO], 2006.

%F a(n) = A121355(n)*n.

%F If A(z) = g.f. of a(n) and B(z) = g.f. of A121355 then A(z) = z d/dz B(z) (Euler operator).

%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,n=1..N),t, ascending);

%t m = 18;

%t s2 = Exp[t + t^2/2] + O[t]^(m+1) // Normal;

%t s3 = Exp[t + t^3/3] + O[t]^(m+1) // Normal;

%t s = Sum[s2[[n+1]] s3[[n+1]]/(t^n/n!), {n, 0, m}];

%t CoefficientList[Log[s] + O[t]^(m+1), t] Range[0, m]! Range[0, m] // Rest (* _Jean-François Alcover_, Sep 02 2018, from Maple *)

%o (PARI) N=18; x='x+O('x^(N+1));

%o A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));

%o A121355_ser = serlaplace(log(serconvol(A121357_ser, exp(x))));

%o Vec(x*A121355_ser') \\ _Gheorghe Coserea_, May 10 2017

%Y Labeled version of A005133.

%Y Labeled and dotted version of A121350.

%Y Dotted version of A121355.

%Y Connected and dotted version of A121357.

%Y Connected, labeled and dotted version of A121352.

%K nonn

%O 1,2

%A _Samuel A. Vidal_, Jul 23 2006