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
a(n) is odd if and only if n is a term of A185647 (this is equivalent to Theorem 2 from Godsil et al.). - Lorenzo Sauras Altuzarra, Mar 03 2026
From Lorenzo Sauras Altuzarra, Apr 07 2026: (Start)
Grady conjectures that: I) if n is odd, then a(n) and a(2*n) have the same parity and II) a(4*n) is even (cf. p. 94 from his article).
If n = p^k > 3 for some integer k > 0 and some prime p, then a(n) == 0 (mod p). This is so because PSL_2(Z) contains no normal subgroups of index n (cf. p. 846 from Newman's 1976 article). (End)
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
C. Godsil, W. Imrich, and R. Razen, On the Number of Subgroups of Given Index in the Modular Group, Monatshefte für Mathematik, vol. 87, 1979, pp. 273-280. See also.
Michael Grady, Counting the subgroups of an infinite group, The Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 20, 1996, pp. 89-96.
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, Mathematics of Computation, vol. 30, 1976, pp. 838-846.
Samuel Alexandre Vidal, Sur la Classification et le Dénombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison, (in French), arXiv:math/0702223 [math.CO], 2007.
FORMULA
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
a(n) = 4*a(n-3) + 2*a(n-4) + (n-3)*a(n-6) - 2*a(n-7) - (n-6)*a(n-9) + Sum_{i=1..n-7} a(n-6-i)*a(i) - Sum_{i=1..n-10} a(n-9-i)*a(i) (this is Theorem 1 from Godsil et al.). - Lorenzo Sauras Altuzarra, Mar 03 2026
a(n) ~ (12*Pi*exp(1/2))^(-1/2)*exp((n/6)*log(n)-n/6+n^(1/2)+n^(1/3)+(1/2)*log(n)) (this is Theorem 4 from the second article by Newman's 1976 article). - Lorenzo Sauras Altuzarra, Apr 07 2026
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
# Alternative:
a := proc(n) option remember: if n < 10 then [1, 1, 4, 8, 5, 22, 42, 40, 120][n]: else 4*a(n-3)+2*a(n-4)+(n-3)*a(n-6)-2*a(n-7)-(n-6)*a(n-9)+add(a(n-6-i)*a(i), i = 1 .. n-7)-add(a(n-9-i)*a(i), i = 1 .. n-10): fi: end:
seq(a(n), n = 1 .. 50); # Lorenzo Sauras Altuzarra, Apr 07 2026
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
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