A121357 Number of different, not necessarily connected, labeled trivalent diagrams of size n.

1, 1, 2, 12, 90, 546, 6156, 81432, 942012, 15114780, 294765336, 5069224656, 108842183352, 2770895886552, 64609245619920, 1742542175582496, 55074355772360976, 1626315165597840912, 53331321825434963232

Offset: 0

Author

Samuel A. Vidal, Jul 23 2006

Keywords

• nonn

Comments

Equivalently, the number of PSL_2(ZZ) actions on a finite labeled set of size n.

Also the number of (r,s) pair of permutations in S_n 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

Unconnected version of A121355.

Labeled version of A121352.

Labeled, unconnected version of A121350.

Cf. also A005133, 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) : sort(add(op(n+1,exs23)*n!,n=0..N),t, ascending);

Mathematica

m = 18; 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}]; CoefficientList[ Sum[exs23[[n + 1]]*n!, {n, 0, m}], t] (* Jean-François Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)

PARI

N=19; x='x+O('x^N);
Vec(serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)))) \\ Gheorghe Coserea, May 10 2017

Formula

If A(z) = g.f. of a(n) and B(z) = g.f. of A121355 then A(z) = exp(B(z)).

Six-term linear recurrence: (n^3 + 12*n^2 + 47*n + 61)*a(n + 6) = (29040 + 239224*n^2 + 127628*n + 20715*n^6 + 252267*n^3 + 166304*n^4 + 71889*n^5 + 33*n^9 + 3943*n^7 + 476*n^8 + n^10)*a(n) + (441*n^4 + 3*n^6 + 2160 + 57*n^5 + 4572*n + 3948*n^2 + 1779*n^3)*a(n + 1) + (34920 + 61314*n + 45886*n^2 + 18989*n^3 + 4697*n^4 + 695*n^5 + 57*n^6 + 2*n^7)*a(n + 2) + (19640 + 79*n^5 + 3*n^6 + 861*n^4 + 27598*n + 16084*n^2 + 4975*n^3)*a(n + 3) + (17*n^3 + 425 + n^4 + 350*n + 113*n^2)*a(n + 4) + (1 + 20*n + 9*n^2 + n^3)*a(n + 5) with n = 0, 1, ...

a(n) = A000085(n) * A001470(n). - Mark van Hoeij, May 13 2013