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

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

Author

Simon Plouffe

Keywords

• nonn
• nice
• easy

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

Crossrefs

Cf. A121357.

Programs

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 *)

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

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

Extensions

More terms from Samuel A. Vidal, Jul 23 2006

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