A027837 Number of subgroups of index n in free group of rank 3.

1, 7, 97, 2143, 68641, 3011263, 173773153, 12785668351, 1169623688353, 130305512589247, 17376934722756577, 2733655173624167551, 501034099176714373921, 105847486567006696384831

Offset

1,2

References

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

M. Hall, Subgroups of finite index in free groups, Canad. J. Math., 1 (1949), 187-190.

V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.

Formula

a(n) = n*n!^2 - Sum_{k=1..n-1} k!^2*a(n-k).

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=1} (n-1)!^2*x^n ). - Paul D. Hanna, Apr 13 2009

Mathematica

a[n_] := a[n] = n*n!^2 - Sum [k!^2*a[n-k], {k, 1, n-1}]; Table[ a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 13 2011, after formula *)

Prog

(PARI) {a(n)=n*polcoeff(log(sum(k=0, n, k!^2*x^k)+x*O(x^n)), n)} \\ Paul D. Hanna, Apr 13 2009
(Haskell)
a027837 n = a027837_list !! (n-1)
a027837_list = f 1 [] where
   f x ys = y : f (x + 1) (y : ys) where
            y = a001044 x * x - sum (zipWith (*) ys $ tail a001044_list)
-- Reinhard Zumkeller, Sep 05 2015

Crossrefs

Cf. A003319, A049290-A049295.

Cf. A001044.

Sequence in context: A367134 A013521 A003710 * A174315 A046908 A005014

Adjacent sequences: A027834 A027835 A027836 * A027838 A027839 A027840

Keyword

easy,nice,nonn

Author

Valery A. Liskovets

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000

Further terms from Naohiro Nomoto, Jun 18 2001

Status

approved