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

Author

Valery A. Liskovets

Keywords

• easy
• nice
• nonn

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.

Crossrefs

Cf. A003319, A049290-A049295.

Cf. A001044.

Programs

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

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 *)

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

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

Extensions

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

Further terms from Naohiro Nomoto, Jun 18 2001