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A049291 Number of subgroups of index n in free group of rank 4. 2
1, 15, 625, 54335, 8563601, 2228419359, 893451975473, 523337983164799, 429463651385469649, 477364501208149290975, 699086688951391180496497, 1318072723102023442664430143, 3137514636520304660660007679505 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 * A092958 A222268 A280179

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

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Last modified July 28 10:57 EDT 2017. Contains 289887 sequences.