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

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

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.

Index entries for sequences related to modular groups

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) is the 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: A276577 A011366 A363873 * A198241 A175475 A193082

Adjacent sequences: A005130 A005131 A005132 * A005134 A005135 A005136

Keyword

nonn,nice,easy

Author

Simon Plouffe

Extensions

More terms from Samuel A. Vidal, Jul 23 2006

Entry revised by N. J. A. Sloane, Jul 25 2006

Status

approved