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

1, 2, 24, 192, 600, 15840, 211680, 1612800, 43545600, 961632000, 11416204800, 365957222400, 10766518963200, 191617884057600, 6758061133824000, 254086360399872000, 6058779650187264000, 241382293453357056000

Offset: 1

Author

Samuel A. Vidal, Jul 23 2006

Keywords

• nonn

Comments

"Dotted" means having a distinguished element. - N. J. A. Sloane, Feb 06 2012

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

Links

• S. A. Vidal, Sur la Classification et le Dénombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison, (in French), arXiv:math/0702223 [math.CO], 2006.

Crossrefs

Labeled version of A005133.

Labeled and dotted version of A121350.

Dotted version of A121355.

Connected and dotted version of A121357.

Connected, labeled and dotted version of A121352.

Programs

Maple

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);

Mathematica

m = 18;
s2 = Exp[t + t^2/2] + O[t]^(m+1) // Normal;
s3 = Exp[t + t^3/3] + O[t]^(m+1) // Normal;
s = Sum[s2[[n+1]] s3[[n+1]]/(t^n/n!), {n, 0, m}];
CoefficientList[Log[s] + O[t]^(m+1), t] Range[0, m]! Range[0, m] // Rest (* Jean-François Alcover, Sep 02 2018, from Maple *)

PARI

N=18; x='x+O('x^(N+1));
A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));
A121355_ser = serlaplace(log(serconvol(A121357_ser, exp(x))));
Vec(x*A121355_ser') \\ Gheorghe Coserea, May 10 2017

Formula

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

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).