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A005133
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Number of index n subgroups of modular group PSL_2(Z).
(Formerly M3320)
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9
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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
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OFFSET
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1,3
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COMMENTS
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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 Alexandre Vidal (samuel.vidal(AT)free.fr), Jul 23 2006
Connected and dotted version of A121352. Dotted version of A121350. Unlabeled version of A121356. Unlabeled and dotted version of A121355. - Samuel Alexandre Vidal (samuel.vidal(AT)free.fr), Jul 23 2006
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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.
Index entries for sequences related to modular groups
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FORMULA
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a(n) = A121355(n)/(n-1)!, a(n) = A121356(n)/n!. - Samuel Alexandre Vidal (samuel.vidal(AT)free.fr), 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 Alexandre Vidal (samuel.vidal(AT)free.fr), Jul 23 2006
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MAPLE
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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 Alexandre Vidal (samuel.vidal(AT)free.fr), Jul 23 2006
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A121357.
Sequence in context: A021677 A124193 A011366 * A198241 A175475 A193082
Adjacent sequences: A005130 A005131 A005132 * A005134 A005135 A005136
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Simon Plouffe
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EXTENSIONS
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More terms from Samuel Alexandre Vidal (samuel.vidal(AT)free.fr), Jul 23 2006
Entry revised by N. J. A. Sloane, Jul 25 2006
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STATUS
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approved
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