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A008307
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Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals.
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0
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1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 3, 2, 1, 1, 26, 9, 4, 1, 1, 1, 76, 21, 16, 1, 2, 1, 1, 232, 81, 56, 1, 6, 1, 1, 1, 764, 351, 256, 25, 18, 1, 2, 1, 1, 2620, 1233, 1072, 145, 66, 1, 4, 1, 1, 1, 9496, 5769, 6224, 505, 396, 1, 16, 3, 2, 1, 1, 35696, 31041, 33616, 1345, 2052, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Solutions to x^k = 1 in Symm_n.
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
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FORMULA
| T(n+1, k) = Sum_{d|k} (n)_{d-1}*T(n-d+1, k), where (n)_i = n*(n-1)*(n-2)*...*(n-i+1).
Sum_{n >= 0} T(n, k)*t^n/n! = exp( sum_{d|k} t^d/d ).
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EXAMPLE
| Array begins
1 1 1 1 1 1 ...
1 2 1 2 1 2 ...
1 4 9 4 1 6 ...
1 10 9 16 1 ...
1 26 21 56 25 ...
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MATHEMATICA
| t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = Sum[(n!/(n - d + 1)!)*t[n - d, k], {d, Divisors[k]}]; t[_, _] = 1; Flatten[ Table[ t[n - k, k], {n, 0, 12}, {k, 1, n}]] (* From Jean-François Alcover, Dec 12 2011, after given formula *)
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CROSSREFS
| Rows give A056595, (more sequences needed!), columns give A000085, A001470, A001472, A052501, A053496-A053505.
Sequence in context: A205553 A178411 A064645 * A192005 A205592 A099238
Adjacent sequences: A008304 A008305 A008306 * A008308 A008309 A008310
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 13 2001
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