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A050213
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Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=5.
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3
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24, 120, 720, 5040, 40320, 362880, 72576, 3628800, 1330560, 39916800, 20338560, 479001600, 303937920, 6227020800, 4643084160, 87178291200, 73721007360, 1743565824, 1307674368000, 1224694598400, 69742632960, 20922789888000
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OFFSET
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5,1
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COMMENTS
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Generalizes Stirling numbers of the first kind.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
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LINKS
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EXAMPLE
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Triangle begins:
05: 24;
06: 120;
07: 720;
08: 5040;
09: 40320;
10: 362880, 72576;
11: 3628800, 1330560;
12: 39916800, 20338560;
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MAPLE
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b:= proc(n) option remember; expand(`if`(n=0, 1, add(
b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=5..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
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MATHEMATICA
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b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*x*Binomial[n - 1, i - 1]* (i - 1)!, {i, 5, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n]];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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