A121355 Number of transitive PSL_2(ZZ) actions on a finite labeled set of size n.

1, 1, 8, 48, 120, 2640, 30240, 201600, 4838400, 96163200, 1037836800, 30496435200, 828193766400, 13686991718400, 450537408921600, 15880397524992000, 356398802952192000, 13410127414075392000, 569542360114151424000, 16614774394239909888000

Offset: 1

Author

Samuel A. Vidal, Jul 23 2006

Keywords

• nonn

Comments

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

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.

Links

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

Crossrefs

Connected version of A121357.

Labeled version of A121350.

Cf. also A005133, A121352, A121356.

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=1..N),t, ascending);
# Alternatively:
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

Mathematica

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 *)

PARI

N=20; x='x+O('x^(N+1));
A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));
Vec(serlaplace(log(serconvol(A121357_ser, exp(x))))) \\ Gheorghe Coserea, May 10 2017

Formula

If A(z) = g.f. of a(n) and B(z) = g.f. of A121357 then A(z) = log(B(z)).