cp's OEIS Frontend

A027837 Number of subgroups of index n in free group of rank 3.

%I A027837 #19 Jun 30 2017 03:01:24
%S A027837 1,7,97,2143,68641,3011263,173773153,12785668351,1169623688353,
%T A027837 130305512589247,17376934722756577,2733655173624167551,
%U A027837 501034099176714373921,105847486567006696384831
%N A027837 Number of subgroups of index n in free group of rank 3.
%D A027837 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
%D A027837 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).
%H A027837 Reinhard Zumkeller, <a href="/A027837/b027837.txt">Table of n, a(n) for n = 1..250</a>
%H A027837 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 A027837 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 A027837 a(n) = n*n!^2 - Sum_{k=1..n-1} k!^2*a(n-k).
%F A027837 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
%t A027837 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 *)
%o A027837 (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
%o A027837 (Haskell)
%o A027837 a027837 n = a027837_list !! (n-1)
%o A027837 a027837_list = f 1 [] where
%o A027837    f x ys = y : f (x + 1) (y : ys) where
%o A027837             y = a001044 x * x - sum (zipWith (*) ys $ tail a001044_list)
%o A027837 -- _Reinhard Zumkeller_, Sep 05 2015
%Y A027837 Cf. A003319, A049290-A049295.
%Y A027837 Cf. A001044.
%K A027837 easy,nice,nonn
%O A027837 1,2
%A A027837 _Valery A. Liskovets_
%E A027837 More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000
%E A027837 Further terms from _Naohiro Nomoto_, Jun 18 2001