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

1,2

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

Table of n, a(n) for n=1..18.

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.

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).

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 *)

Prog

(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

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.

Sequence in context: A131972 A059387 A126190 * A052780 A245019 A189769

Adjacent sequences: A121353 A121354 A121355 * A121357 A121358 A121359

Keyword

nonn

Author

Samuel A. Vidal, Jul 23 2006

Status

approved