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

0,3

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

Table of n, a(n) for n=0..18.

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.

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

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 *)

Prog

(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

Crossrefs

Unconnected version of A121355.

Labeled version of A121352.

Labeled, unconnected version of A121350.

Cf. also A005133, A121356.

Sequence in context: A224152 A174356 A355138 * A098926 A074610 A372109

Adjacent sequences: A121354 A121355 A121356 * A121358 A121359 A121360

Keyword

nonn

Author

Samuel A. Vidal, Jul 23 2006

Status

approved