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A171998
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In general, let A(n,k,m) denote the (n,k)-th entry of the inverse of the matrix consisting of the (n,k)-th m-restrained Stirling numbers of the second kind (-1)^(n-k) times the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, as the (n+1,k+1)-th entry. The sequence shows A(n,k,3), which is a lower triangular matrix, read by rows.
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1
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1, 1, 1, 1, 3, 1, -5, 7, 6, 1, -65, -15, 25, 10, 1, -455, -455, 0, 65, 15, 1, -1295, -4725, -1715, 140, 140, 21, 1
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OFFSET
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1,5
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COMMENTS
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A(n,k,m) also can be expanded for nonpositive integers n and k using the m-restrained Stirling numbers of the first kind.
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LINKS
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FORMULA
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A(n,k,m) = A(n-1,k-1,m) - Sum_{i=1..m-1} (-1)^{i}(k)...(k+i-1) A(n,k+i,m) A(n,k,m) = A(n-1,k-1,m) + k A(n-1,k,m) + (-1)^m k(k+1)...(k+m-1)A(n,k+m,m).
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EXAMPLE
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A(1,1,3) = 1, A(1,2,3) = 0, A(1,3,3) = 0, A(1,4,3) = 0, ...
A(2,1,3) = 1, A(2,2,3) = 1, A(2,3,3) = 0, A(2,4,3) = 0, ...
A(3,1,3) = 1, A(3,2,3) = 3, A(3,3,3) = 1, A(3,4,3) = 0, ...
A(4,1,3) = -5, A(4,2,3) = 7, A(4,3,3) = 6, A(4,4,3) = 1, ...
In other words, A(n,k,3) is the matrix
1
1 1
1 3 1
-5 7 6 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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