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A005133 Number of index n subgroups of modular group PSL_2(Z).
(Formerly M3320)
9

%I M3320

%S 1,1,4,8,5,22,42,40,120,265,286,764,1729,2198,5168,12144,17034,37702,

%T 88958,136584,288270,682572,1118996,2306464,5428800,9409517,19103988,

%U 44701696,80904113,163344502,379249288,711598944,1434840718,3308997062,6391673638,12921383032,29611074174,58602591708,119001063028,271331133136,547872065136,1119204224666,2541384297716,5219606253184,10733985041978,24300914061436,50635071045768,104875736986272,236934212877684,499877970985660

%N Number of index n subgroups of modular group PSL_2(Z).

%C Equivalently, the number of isomorphism class of transitive PSL_2(Z) actions on a finite dotted (i.e. having a distinguished element) set of size n. Also the number of different connected dotted trivalent diagrams of size n. - _Samuel A. Vidal_, Jul 23 2006

%C Connected and dotted version of A121352. Dotted version of A121350. Unlabeled version of A121356. Unlabeled and dotted version of A121355. - _Samuel A. Vidal_, Jul 23 2006

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A005133/b005133.txt">Table of n, a(n) for n = 1..1000</a>

%H Morris Newman, <a href="https://doi.org/10.1090/S0002-9947-1967-0204375-3">Classification of Normal Subgroups of the Modular Group</a>, Transactions of the American Mathematical Society 126 (1967), no. 2, 267-277.

%H Morris Newman, <a href="http://dx.doi.org/10.1090/S0025-5718-1976-0466047-9">Asymptotic formulas related to free products of cyclic groups</a>, Math. Comp. 30 (1976), no. 136, 838-846.

%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], 2007.

%H <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>

%F a(n) = A121355(n)/(n-1)!, a(n) = A121356(n)/n!. - _Samuel A. Vidal_, Jul 23 2006

%F If A(z) = g.f. of a(n) and B(z) = g.f. of A121356 then A(z) = Borel transform of B(z). - _Samuel A. Vidal_, Jul 23 2006

%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) : logexs23:=sort(convert(taylor(log(exs23),t,N+1),polynom),t, ascending) : sort(add(op(n,logexs23)*n,n=1..N),t, ascending) ; # _Samuel A. Vidal_, Jul 23 2006

%t m = 50; 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+1}] // Normal; CoefficientList[ Sum[ logexs23[[n]]*n, {n, 1, m}], t] // Rest (* _Jean-Fran├žois Alcover_, Dec 05 2012, translated from Maple *)

%o (PARI) N=50; x='x+O('x^(N+1));

%o A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));

%o Vec(x*log(serconvol(A121357_ser, exp(x)))') \\ _Gheorghe Coserea_, May 10 2017

%Y Cf. A121357.

%K nonn,nice,easy

%O 1,3

%A _Simon Plouffe_

%E More terms from _Samuel A. Vidal_, Jul 23 2006

%E Entry revised by _N. J. A. Sloane_, Jul 25 2006

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Last modified November 13 19:25 EST 2018. Contains 317149 sequences. (Running on oeis4.)