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A224211
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Irregular triangular array read by rows. T(n,k) is the number of n-permutations with exactly k distinct cycle lengths; n>=1, 1<=k<=floor( (-1+(1+8n)^(1/2))/2 ).
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1
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1, 1, 2, 3, 6, 8, 24, 50, 120, 234, 120, 720, 1764, 630, 5040, 11808, 7392, 40320, 109584, 69552, 362880, 954000, 763200, 151200, 3628800, 10628640, 8165520, 1330560, 39916800, 113891040, 109010880, 25280640, 479001600, 1486442880, 1345687200, 381775680, 6227020800, 18913184640, 19773804960, 6763236480, 87178291200, 283465647360, 291950568000, 102508005600, 10897286400
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OFFSET
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1,3
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COMMENTS
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Row sums = A007838.
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LINKS
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Alois P. Heinz, Rows n = 1..300, flattened
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FORMULA
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E.g.f.: Product_{i>=1} (1 + y*x)^i/i.
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EXAMPLE
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: 1;
: 1;
: 2, 3;
: 6, 8;
: 24, 50;
: 120, 234, 120;
: 720, 1764, 630;
: 5040, 11808, 7392;
: 40320, 109584, 69552;
: 362880, 954000, 763200, 151200;
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MAPLE
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b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+ (i-1)!*b(n-i, i-1)*
`if`(i>n, 0, binomial(n, i)*x))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..15); # Alois P. Heinz, Oct 21 2015
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MATHEMATICA
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nn=15; f[list_]:=Select[list, #>0&]; Map[f, Drop[Range[0, nn]!CoefficientList[Series[Product[(1+y x^i/i), {i, 1, nn}], {x, 0, nn}], {x, y}], 1]]//Grid
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CROSSREFS
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Sequence in context: A093705 A281645 A351853 * A187026 A179221 A350119
Adjacent sequences: A224208 A224209 A224210 * A224212 A224213 A224214
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KEYWORD
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nonn,tabf
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AUTHOR
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Geoffrey Critzer, Apr 01 2013
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STATUS
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approved
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