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A049290
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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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8
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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; internal format)
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OFFSET
| 1,5
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REFERENCES
| 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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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 1..703
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.
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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, ...
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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
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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}]] (* From Jean-François Alcover, Nov 09 2011, after Maple *)
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CROSSREFS
| Rows give A003319, A027837, A049291, columns give A000225, A049294, A049295. Main diagonal is A057014.
Sequence in context: A075440 A137470 A112492 * A147990 A134567 A131932
Adjacent sequences: A049287 A049288 A049289 * A049291 A049292 A049293
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KEYWORD
| nonn,easy,nice,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 09 2000
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EXTENSIONS
| More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 29 2009
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