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A173333
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Triangle read by rows: T(n,k) = n! / k!, 1<=k<=n.
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30
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1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 120, 60, 20, 5, 1, 720, 360, 120, 30, 6, 1, 5040, 2520, 840, 210, 42, 7, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 3628800, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1
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OFFSET
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1,2
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COMMENTS
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1 < k <= n: T(n,k) = T(n,k-1) / k;
1 <= k <= n: T(n+1,k) = A119741(n,n-k+1);
1 <= k <= n: T(n+1,k+1) = A162995(n,k);
T(n-1,k), k=1,...,n-1, gives the number of representative necklaces with n beads (C_N symmetry) of n+1-k distinct colors, say c[1],c[2],...,c[n-k+1], corresponding to the color signature determined by the partition k,1^(n-k) of n. The representative necklaces have k beads of color c[1]. E.g., n=4, k=2: partition 2,1,1, color signature (parts as exponents) c[1]c[1]c[2]c[3], 3=T(3,2) necklaces (write j for color c[j]): cyclic(1123), cyclic(1132) and cyclic(1213). See A212359 for the numbers for general partitions or color signatures. (End)
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LINKS
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FORMULA
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EXAMPLE
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Triangle starts:
n\k 1 2 3 4 5 6 7 8 9 10 ...
1 1
2 2 1
3 6 3 1
4 24 12 4 1
5 120 60 20 5 1
6 720 360 120 30 6 1
7 5040 2520 840 210 42 7 1
8 40320 20160 6720 1680 336 56 8 1
9 362880 181440 60480 15120 3024 504 72 9 1
10 3628800 1814400 604800 151200 30240 5040 720 90 10 1
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MATHEMATICA
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PROG
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(Haskell)
a173333 n k = a173333_tabl !! (n-1) !! (k-1)
a173333_row n = a173333_tabl !! (n-1)
a173333_tabl = map fst $ iterate f ([1], 2)
where f (row, i) = (map (* i) row ++ [1], i + 1)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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