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A027837
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Number of subgroups of index n in free group of rank 3.
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7
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1, 7, 97, 2143, 68641, 3011263, 173773153, 12785668351, 1169623688353, 130305512589247, 17376934722756577, 2733655173624167551, 501034099176714373921, 105847486567006696384831
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| M. Hall, Subgroups of finite index in free groups, Canad. J. Math., 1 (1949), 187-190.
P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
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.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).
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FORMULA
| a(n)=n*n!^2-Sum k!^2*a(n-k), k=1..n-1.
L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=1} (n-1)!^2*x^n ). [From Paul D. Hanna (pauldhanna(AT)juno.com), Apr 13 2009]
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MATHEMATICA
| a[n_] := a[n] = n*n!^2 - Sum [k!^2*a[n-k], {k, 1, n-1}]; Table[ a[n], {n, 1, 14}] (* From Jean-François Alcover, Dec 13 2011, after formula *)
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PROG
| (PARI) {a(n)=n*polcoeff(log(sum(k=0, n, k!^2*x^k)+x*O(x^n)), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Apr 13 2009]
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CROSSREFS
| Cf. A003319, A049290-A049295.
Sequence in context: A132061 A013521 A003710 * A174315 A046908 A005014
Adjacent sequences: A027834 A027835 A027836 * A027838 A027839 A027840
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KEYWORD
| easy,nice,nonn
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AUTHOR
| V. A. Liskovets (liskov(AT)im.bas-net.by)
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000
Further terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), Jun 18 2001
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