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A211603
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Triangular array read by rows: T(n,k) is the number of n-permutations that are pure cycles having exactly k fixed points; n>=2, 0<=k<=n-2.
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4
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1, 2, 3, 6, 8, 6, 24, 30, 20, 10, 120, 144, 90, 40, 15, 720, 840, 504, 210, 70, 21, 5040, 5760, 3360, 1344, 420, 112, 28, 40320, 45360, 25920, 10080, 3024, 756, 168, 36, 362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45, 3628800, 3991680, 2217600, 831600, 237600, 55440, 11088, 1980, 330, 55
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OFFSET
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2,2
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COMMENTS
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Equivalently, T(n,k) is the number of n-permutations that are pure cycles of length n-k.
With a different row and column indexing, this triangle equals the infinitesimal generator of A008290. Equals the unsigned version of A238363, omitting its main diagonal. See also A092271. - Peter Bala, Feb 13 2017
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LINKS
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FORMULA
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E.g.f.: exp(y*x)*(log(1/(1-x))-x).
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EXAMPLE
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T(3,1) = 3 because we have (1)(2,3), (2)(1,3), (3)(1,2).
1;
2, 3;
6, 8, 6;
24, 30, 20, 10;
120, 144, 90, 40, 15;
720, 840, 504, 210, 70, 21;
5040, 5760, 3360, 1344, 420, 112, 28;
40320, 45360, 25920, 10080, 3024, 756, 168, 36;
362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45;
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MAPLE
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T:= (n, k)-> binomial(n, k)*(n-k-1)!:
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MATHEMATICA
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nn=10; f[list_]:=Select[list, #>0&]; Map[f, Range[0, nn]!CoefficientList[ Series[Exp[y x](Log[1/(1-x)]-x), {x, 0, nn}], {x, y}]]//Grid
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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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