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A027837 Number of subgroups of index n in free group of rank 3. 9
1, 7, 97, 2143, 68641, 3011263, 173773153, 12785668351, 1169623688353, 130305512589247, 17376934722756577, 2733655173624167551, 501034099176714373921, 105847486567006696384831 (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
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. A001044.
Sequence in context: A367134 A013521 A003710 * A174315 A046908 A005014
KEYWORD
easy,nice,nonn
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000
Further terms from Naohiro Nomoto, Jun 18 2001
STATUS
approved

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Last modified March 28 08:12 EDT 2024. Contains 371236 sequences. (Running on oeis4.)