%I #22 Feb 04 2023 09:45:52
%S 24,120,720,5040,40320,362880,72576,3628800,1330560,39916800,20338560,
%T 479001600,303937920,6227020800,4643084160,87178291200,73721007360,
%U 1743565824,1307674368000,1224694598400,69742632960,20922789888000
%N Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=5.
%C Generalizes Stirling numbers of the first kind.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
%H Alois P. Heinz, <a href="/A050213/b050213.txt">Rows n = 5..300, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationCycle.html">Permutation Cycle.</a>
%e Triangle begins:
%e 05: 24;
%e 06: 120;
%e 07: 720;
%e 08: 5040;
%e 09: 40320;
%e 10: 362880, 72576;
%e 11: 3628800, 1330560;
%e 12: 39916800, 20338560;
%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=5..n)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p seq(T(n), n=5..20); # _Alois P. Heinz_, Sep 25 2016
%t 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 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n]];
%t T /@ Range[5, 20] // Flatten (* _Jean-François Alcover_, Dec 08 2019, after _Alois P. Heinz_ *)
%Y Cf. A008275, A008306, A050211, A050212.
%K nonn,tabf
%O 5,1
%A _Eric W. Weisstein_
%E Offset changed from 1 to 5 by _Alois P. Heinz_, Sep 25 2016
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