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%I #14 Jun 30 2017 03:01:51
%S 1,15,625,54335,8563601,2228419359,893451975473,523337983164799,
%T 429463651385469649,477364501208149290975,699086688951391180496497,
%U 1318072723102023442664430143,3137514636520304660660007679505
%N Number of subgroups of index n in free group of rank 4.
%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).
%H M. Hall, <a href="http://dx.doi.org/10.4153/CJM-1949-017-2">Subgroups of finite index in free groups</a>, Canad. J. Math., 1 (1949), 187-190.
%H V. A. Liskovets and A. Mednykh, <a href="http://dx.doi.org/10.1080/00927870008826924">Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces</a>, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
%F a(n) = n*n!^3 - Sum_{k=1..n-1} k!^3*a(n-k).
%F 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]
%t 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 *)
%o (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
%Y Cf. A003319, A027837, A049290, A049292, A049293, A049294, A049295.
%K easy,nice,nonn
%O 1,2
%A _Valery A. Liskovets_
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
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