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

1,2

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

Table of n, a(n) for n=1..13.

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!^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]

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 *)

Prog

(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

Crossrefs

Cf. A003319, A027837, A049290, A049292, A049293, A049294, A049295.

Sequence in context: A012210 A203172 A081022 * A351180 A092958 A222268

Adjacent sequences: A049288 A049289 A049290 * A049292 A049293 A049294

Keyword

easy,nice,nonn

Author

Valery A. Liskovets

Extensions

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

Status

approved