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A293211 Triangle T(n,k) is the number of permutations on n elements with at least one k-cycle for 1 <= k <= n.

%I A293211 #21 Oct 12 2017 14:14:29
%S A293211 1,1,1,4,3,2,15,9,8,6,76,45,40,30,24,455,285,200,180,144,120,3186,
%T A293211 1995,1400,1260,1008,840,720,25487,15855,11200,8820,8064,6720,5760,
%U A293211 5040,229384,142695,103040,79380,72576,60480,51840,45360,40320,2293839,1427895,1030400,793800,653184,604800,518400,453600,403200,362880
%N A293211 Triangle T(n,k) is the number of permutations on n elements with at least one k-cycle for 1 <= k <= n.
%C A293211 T(n,k) is equivalent to n! minus the number of permutations on n elements with zero k-cycles (sequence A122974).
%H A293211 Dennis P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/NOKCYCLB.pdf">The number of permutations with no k-cycles</a>
%F A293211 T(n,k) = n! * Sum_{j=1..floor(n/k)} (-1)^(j+1)*(1/k)^j/j!.
%F A293211 T(n,k) = n! - A122974(n,k).
%F A293211 E.g.f. of column k: (1-exp(-x^k/k))/(1-x). - _Alois P. Heinz_, Oct 11 2017
%e A293211 T(n,k) (the first 8 rows):
%e A293211 :     1;
%e A293211 :     1,     1;
%e A293211 :     4,     3,     2;
%e A293211 :    15,     9,     8,    6;
%e A293211 :    76,    45,    40,   30,   24;
%e A293211 :   455,   285,   200,  180,  144,  120;
%e A293211 :  3186,  1995,  1400, 1260, 1008,  840,  720;
%e A293211 : 25487, 15855, 11200, 8820, 8064, 6720, 5760, 5040;
%e A293211   ...
%e A293211 T(4,3)=8 since there are exactly 8 permutations on {1,2,3,4} with at least one 3-cycle: (1)(234), (1)(243), (2)(134), (2)(143), (3)(124), (3)(142), (4)(123), and (4)(132).
%p A293211 T:=(n,k)->n!*sum((-1)^(j+1)*(1/k)^j/j!,j=1..floor(n/k)); seq(seq(T(n,k),k=1..n),n=1..10);
%t A293211 Table[n!*Sum[(-1)^(j + 1)*(1/k)^j/j!, {j, Floor[n/k]}], {n, 10}, {k, n}] // Flatten (* _Michael De Vlieger_, Oct 02 2017 *)
%Y A293211 Columns k=1-10 give: A002467, A027616, A027617, A029571, A029572, A029573, A029574, A029575, A029576, A029577.
%Y A293211 Row sums give A132961.
%Y A293211 T(n,n) gives A000142(n-1) for n>0.
%Y A293211 T(2n,n) gives A052145.
%Y A293211 Cf. A122974, A126074.
%K A293211 easy,nonn,tabl
%O A293211 1,4
%A A293211 _Dennis P. Walsh_, Oct 02 2017