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%I #33 Sep 04 2021 19:17:22
%S 1,1,2,12,90,546,6156,81432,942012,15114780,294765336,5069224656,
%T 108842183352,2770895886552,64609245619920,1742542175582496,
%U 55074355772360976,1626315165597840912,53331321825434963232
%N Number of different, not necessarily connected, labeled trivalent diagrams of size n.
%C Equivalently, the number of PSL_2(ZZ) actions on a finite labeled set of size n.
%C 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.
%H S. A. Vidal, <a href="https://arxiv.org/abs/math/0702223">Sur la Classification et le Denombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison</a>, (in French), arXiv:math/0702223 [math.CO] 2006.
%F If A(z) = g.f. of a(n) and B(z) = g.f. of A121355 then A(z) = exp(B(z)).
%F 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, ...
%F a(n) = A000085(n) * A001470(n). - _Mark van Hoeij_, May 13 2013
%p 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);
%t 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 *)
%o (PARI) N=19; x='x+O('x^N);
%o Vec(serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)))) \\ _Gheorghe Coserea_, May 10 2017
%Y Unconnected version of A121355.
%Y Labeled version of A121352.
%Y Labeled, unconnected version of A121350.
%Y Cf. also A005133, A121356.
%K nonn
%O 0,3
%A _Samuel A. Vidal_, Jul 23 2006
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