cp's OEIS Frontend

A121356 Number of transitive PSL_2(ZZ) actions on a finite dotted and labeled set of size n.

%I A121356 #23 Oct 29 2018 07:16:20
%S A121356 1,2,24,192,600,15840,211680,1612800,43545600,961632000,11416204800,
%T A121356 365957222400,10766518963200,191617884057600,6758061133824000,
%U A121356 254086360399872000,6058779650187264000,241382293453357056000
%N A121356 Number of transitive PSL_2(ZZ) actions on a finite dotted and labeled set of size n.
%C A121356 "Dotted" means having a distinguished element. - _N. J. A. Sloane_, Feb 06 2012
%C A121356 Equivalently, the number of different connected, dotted and labeled trivalent diagrams of size n.
%H A121356 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 A121356 a(n) = A121355(n)*n.
%F A121356 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 A121356 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 A121356 m = 18;
%t A121356 s2 = Exp[t + t^2/2] + O[t]^(m+1) // Normal;
%t A121356 s3 = Exp[t + t^3/3] + O[t]^(m+1) // Normal;
%t A121356 s = Sum[s2[[n+1]] s3[[n+1]]/(t^n/n!), {n, 0, m}];
%t A121356 CoefficientList[Log[s] + O[t]^(m+1), t] Range[0, m]! Range[0, m] // Rest (* _Jean-François Alcover_, Sep 02 2018, from Maple *)
%o A121356 (PARI) N=18; x='x+O('x^(N+1));
%o A121356 A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));
%o A121356 A121355_ser = serlaplace(log(serconvol(A121357_ser, exp(x))));
%o A121356 Vec(x*A121355_ser') \\ _Gheorghe Coserea_, May 10 2017
%Y A121356 Labeled version of A005133.
%Y A121356 Labeled and dotted version of A121350.
%Y A121356 Dotted version of A121355.
%Y A121356 Connected and dotted version of A121357.
%Y A121356 Connected, labeled and dotted version of A121352.
%K A121356 nonn
%O A121356 1,2
%A A121356 _Samuel A. Vidal_, Jul 23 2006