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Array T(n,k) = number of subgroups of index k in free group of rank n, read by antidiagonals.
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%I #26 Jun 05 2015 09:22:17

%S 1,1,1,1,3,1,1,7,13,1,1,15,97,71,1,1,31,625,2143,461,1,1,63,3841,

%T 54335,68641,3447,1,1,127,23233,1321471,8563601,3011263,29093,1,1,255,

%U 139777,31817471,1035045121,2228419359,173773153,273343,1,1,511,839425

%N Array T(n,k) = number of subgroups of index k in free group of rank n, read by antidiagonals.

%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.

%D J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

%H Alois P. Heinz, <a href="/A049290/b049290.txt">Antidiagonals n = 1..37, flattened</a>

%H J. H. Kwak and J. Lee, <a href="http://com2mac.postech.ac.kr/resorce/Lecture_text.htm">Enumeration of graph coverings and surface branched coverings</a>, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3. [Broken link?]

%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.

%e Array T(n,k) (n >= 1, k >= 1) begins:

%e 1, 1, 1, 1, 1, ...

%e 1, 3, 13, 71, 461, ...

%e 1, 7, 97, 2143, 68641, ...

%e 1, 15, 625, 54335, 8563601, ...

%p T:= proc(n,k) option remember; k* k!^(n-1) -add(j!^(n-1) *T(n, k-j), j=1..k-1) end: seq(seq(T(d+1-k, k), k=1..d), d=1..10); # _Alois P. Heinz_, Oct 29 2009

%t nmax = 10; t[n_, k_] := t[n, k] = k*k!^(n-1) - Sum[j!^(n-1)*t[n, k-j], {j, 1, k-1}]; Flatten[ Table[ t[n-k+1, k], {n, 1, nmax}, {k, 1, n}]] (* _Jean-François Alcover_, Nov 09 2011, after Maple *)

%Y Rows give A003319, A027837, A049291.

%Y Columns give A000225, A049294, A049295.

%Y Main diagonal is A057014.

%K nonn,easy,nice,tabl

%O 1,5

%A _N. J. A. Sloane_, Sep 09 2000

%E More terms from _Alois P. Heinz_, Oct 29 2009