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A090657
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Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).
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9
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1, 0, 1, 0, 2, 2, 0, 3, 18, 6, 0, 4, 84, 144, 24, 0, 5, 300, 1500, 1200, 120, 0, 6, 930, 10800, 23400, 10800, 720, 0, 7, 2646, 63210, 294000, 352800, 105840, 5040, 0, 8, 7112, 324576, 2857680, 7056000, 5362560, 1128960, 40320
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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LINKS
| Alois P. Heinz, Rows n = 0..62, flattened
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FORMULA
| T(n,k) = C(n,k) * k! * A048993(n,k).
T(n,k) = A008279(n,k) * A048993(n,k).
T(n,k) = C(n,k) * A019538(n, k).
T(n,k) = C(n,k) * Sum_{j=0..k} (-1)^(k-j) * C(k,j) * j^n.
T(n,k) = n * (T(n-1,k-1) + k/(n-k) * T(n-1,k)) with T(n,n) = n! and T(n,0) = 0 for n>0.
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EXAMPLE
| Triangle begins:
1;
0, 1;
0, 2, 2;
0, 3, 18, 6;
0, 4, 84, 144, 24;
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MAPLE
| T:= proc(n, k) option remember;
if k=n then n!
elif k=0 or k>n then 0
else n * (T(n-1, k-1) + k/(n-k) * T(n-1, k))
fi
end:
seq (seq (T(n, k), k=0..n), n=0..10);
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MATHEMATICA
| Table[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 0, n}], {n, 0, 10}] // Flatten *Geoffrey Critzer, September 9, 2011*
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CROSSREFS
| Row sums give: A000312. Columns k=0-2 give: A000007, A001477, A068605. Diagonal, lower diagonal give: A000142, A001804. Cf. A007318, A048993, A019538, A008279.
Sequence in context: A065484 A011137 A143396 * A167001 A108563 A138476
Adjacent sequences: A090654 A090655 A090656 * A090658 A090659 A090660
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 14 2003
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EXTENSIONS
| Revised description from Jan Maciak, Apr 25 2004
Edited by Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jan 17 2011
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