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A049290 Array T(n,k) = number of subgroups of index k in free group of rank n, read by antidiagonals. 8
1, 1, 1, 1, 3, 1, 1, 7, 13, 1, 1, 15, 97, 71, 1, 1, 31, 625, 2143, 461, 1, 1, 63, 3841, 54335, 68641, 3447, 1, 1, 127, 23233, 1321471, 8563601, 3011263, 29093, 1, 1, 255, 139777, 31817471, 1035045121, 2228419359, 173773153, 273343, 1, 1, 511, 839425 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
REFERENCES
P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
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.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).
LINKS
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3. [Broken link?]
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.
EXAMPLE
Array T(n,k) (n >= 1, k >= 1) begins:
1, 1, 1, 1, 1, ...
1, 3, 13, 71, 461, ...
1, 7, 97, 2143, 68641, ...
1, 15, 625, 54335, 8563601, ...
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Rows give A003319, A027837, A049291.
Columns give A000225, A049294, A049295.
Main diagonal is A057014.
Sequence in context: A112492 A210574 A353532 * A297191 A147990 A134567
KEYWORD
nonn,easy,nice,tabl
AUTHOR
N. J. A. Sloane, Sep 09 2000
EXTENSIONS
More terms from Alois P. Heinz, Oct 29 2009
STATUS
approved

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Last modified May 29 18:36 EDT 2024. Contains 372952 sequences. (Running on oeis4.)