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 A005133 Number of index n subgroups of modular group PSL_2(Z). (Formerly M3320) 9
 1, 1, 4, 8, 5, 22, 42, 40, 120, 265, 286, 764, 1729, 2198, 5168, 12144, 17034, 37702, 88958, 136584, 288270, 682572, 1118996, 2306464, 5428800, 9409517, 19103988, 44701696, 80904113, 163344502, 379249288, 711598944, 1434840718, 3308997062, 6391673638, 12921383032, 29611074174, 58602591708, 119001063028, 271331133136, 547872065136, 1119204224666, 2541384297716, 5219606253184, 10733985041978, 24300914061436, 50635071045768, 104875736986272, 236934212877684, 499877970985660 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 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 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Morris Newman, Classification of Normal Subgroups of the Modular Group, Transactions of the American Mathematical Society 126 (1967), no. 2, 267-277. Morris Newman, Asymptotic formulas related to free products of cyclic groups, Math. Comp. 30 (1976), no. 136, 838-846. 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], 2007. FORMULA a(n) = A121355(n)/(n-1)!, a(n) = A121356(n)/n!. - Samuel A. Vidal, Jul 23 2006 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 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) ; # Samuel A. Vidal, Jul 23 2006 MATHEMATICA 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-François Alcover, Dec 05 2012, translated from Maple *) PROG (PARI) N=50; x='x+O('x^(N+1)); A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3))); Vec(x*log(serconvol(A121357_ser, exp(x)))') \\ Gheorghe Coserea, May 10 2017 CROSSREFS Cf. A121357. Sequence in context: A124193 A276577 A011366 * A198241 A175475 A193082 Adjacent sequences:  A005130 A005131 A005132 * A005134 A005135 A005136 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS More terms from Samuel A. Vidal, Jul 23 2006 Entry revised by N. J. A. Sloane, Jul 25 2006 STATUS approved

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Last modified October 15 18:26 EDT 2019. Contains 328037 sequences. (Running on oeis4.)