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A050213 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=5. 3
24, 120, 720, 5040, 40320, 362880, 72576, 3628800, 1330560, 39916800, 20338560, 479001600, 303937920, 6227020800, 4643084160, 87178291200, 73721007360, 1743565824, 1307674368000, 1224694598400, 69742632960, 20922789888000 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

5,1

COMMENTS

Generalizes Stirling numbers of the first kind

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.

LINKS

Alois P. Heinz, Rows n = 5..300, flattened

Eric Weisstein's World of Mathematics, Permutation Cycle.

EXAMPLE

Triangle begins:

05:       24;

06:      120;

07:      720;

08:     5040;

09:    40320;

10:   362880,    72576;

11:  3628800,  1330560;

12: 39916800, 20338560;

MAPLE

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)):

seq(T(n), n=5..20);  # Alois P. Heinz, Sep 25 2016

MATHEMATICA

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]];

T /@ Range[5, 20] // Flatten (* Jean-Fran├žois Alcover, Dec 08 2019, after Alois P. Heinz *)

CROSSREFS

Cf. A008275, A008306, A050211, A050212.

Sequence in context: A179720 A235702 A052754 * A124657 A342856 A293050

Adjacent sequences:  A050210 A050211 A050212 * A050214 A050215 A050216

KEYWORD

nonn,tabl

AUTHOR

Eric W. Weisstein

EXTENSIONS

Offset changed from 1 to 5 by Alois P. Heinz, Sep 25 2016

STATUS

approved

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Last modified September 26 14:45 EDT 2021. Contains 347668 sequences. (Running on oeis4.)