A049291 Number of subgroups of index n in free group of rank 4.

1, 15, 625, 54335, 8563601, 2228419359, 893451975473, 523337983164799, 429463651385469649, 477364501208149290975, 699086688951391180496497, 1318072723102023442664430143, 3137514636520304660660007679505

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

• 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, A027837, A049290, A049292, A049293, A049294, A049295.

Programs

Mathematica

ClearAll[a]; a[n_] := a[n] = n*n!^3 - Sum [k!^3*a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 13}]  (* Jean-François Alcover, Oct 08 2012, from first formula *)

PARI

{a(n)=n*polcoeff(log(sum(k=0,n,k!^3*x^k)+x*O(x^n)),n)} \\ Paul D. Hanna, Apr 13 2009

Formula

a(n) = n*n!^3 - Sum_{k=1..n-1} k!^3*a(n-k).

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=1} (n-1)!^3*x^n ). [Paul D. Hanna, Apr 13 2009]

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001