A057012 -id:A057012 - cp's OEIS frontend

A057013 Number of conjugacy classes of subgroups of index n in free group of rank n.

1, 3, 41, 14120, 207009649, 193466859054994, 16390021452000128467345, 173238117954686651535535048940928, 300679800156605267097630253972085058107938561

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
Index entries for sequences related to groups

Cf. A057004-A057012.

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank n.

Original entry on oeis.org

1, 7, 41, 235, 1361, 7987, 47321, 281995, 1685921, 10096867, 60524201, 362972155, 2177309681, 13062280147, 78368930681, 470199300715, 2821152888641, 16926788453827, 101560343826761, 609360901747675, 3656161925798801, 21936961098633907, 131621735219132441

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

Starting at a(2), consider that 2/3 - 1/2 = 1/6 with 1+6=7=a(2); 8/9 - 3/4 = 5/36 with 5+36=41=a(3); 26/27 - 7/8=19/216 with 19+216=235=a(4); 80/81 - 15/16=65/1296 with 65+1296=1361=a(5) and so forth. The numerators starting at a(3) are 5,19,65,211,665,2059,6305,... (see A001047) with 19 mod 5=4, 65 mod 19=8, 211 mod 65=16, 665 mod 211=32, 2059 mod 665=64, 6305 mod 2059=128, and so forth for higher powers of 2. - J. M. Bergot, May 09 2015

In other words, let f(n) = (3^(n-1)-1)/3^(n-1) - (2^(n-1)-1)/2^(n-1), then for n>=1 a(n) = numerator(f(n)) + denominator(f(n)). - Michel Marcus, May 29 2015

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Cf. A057004, A057005, A057006, A057007, A057008, A057010, A057011, A057012, A057013.

[6^(n-1)+3^(n-1)-2^(n-1): n in [1..30]] /* or */ I:=[1,7,41]; [n le 3 select I[n] else 11*Self(n-1)-36*Self(n-2)+36*Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 12 2015

Table[6^(n-1)+3^(n-1)-2^(n-1),{n,25}] (* or *) LinearRecurrence[ {11,-36,36},{1,7,41},25] (* Harvey P. Dale, Nov 24 2011 *)
CoefficientList[Series[(1 - 4 x)/((1 - 2 x) (1 - 3 x) (1 - 6 x)), {x, 0, 33}], x] (* Vincenzo Librandi, May 12 2015 *)

PARI
```
a(n)=if(n<0,0,6^(n-1)+3^(n-1)-2^(n-1))
```

G.f.: x(1-4x)/((1-2x)(1-3x)(1-6x)). a(n) = 6^(n-1)+3^(n-1)-2^(n-1).
E.g.f.: e^(6*x)+e^(3*x)-e^(2*x). [Mohammad K. Azarian, Jan 16 2009]
a(1)=1, a(2)=7, a(3)=41, a(n) = 11*a(n-1)-36*a(n-2)+36*a(n-3). [Harvey P. Dale, Nov 24 2011]

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

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A057013 Number of conjugacy classes of subgroups of index n in free group of rank n.

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A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank n.

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