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A211603 Triangular array read by rows: T(n,k) is the number of n-permutations that are pure cycles having exactly k fixed points; n>=2, 0<=k<=n-2.

%I A211603 #26 Feb 14 2017 02:38:13
%S A211603 1,2,3,6,8,6,24,30,20,10,120,144,90,40,15,720,840,504,210,70,21,5040,
%T A211603 5760,3360,1344,420,112,28,40320,45360,25920,10080,3024,756,168,36,
%U A211603 362880,403200,226800,86400,25200,6048,1260,240,45,3628800,3991680,2217600,831600,237600,55440,11088,1980,330,55
%N A211603 Triangular array read by rows: T(n,k) is the number of n-permutations that are pure cycles having exactly k fixed points; n>=2, 0<=k<=n-2.
%C A211603 Equivalently, T(n,k) is the number of n-permutations that are pure cycles of length n-k.
%C A211603 Row sums = A006231.
%C A211603 With a different row and column indexing, this triangle equals the infinitesimal generator of A008290. Equals the unsigned version of A238363, omitting its main diagonal. See also A092271. - _Peter Bala_, Feb 13 2017
%H A211603 Alois P. Heinz, <a href="/A211603/b211603.txt">Rows n = 2..150, flattened</a>
%F A211603 E.g.f.: exp(y*x)*(log(1/(1-x))-x).
%F A211603 T(n,k) = C(n,k)*(n-k-1)!. - _Alois P. Heinz_, Feb 10 2013
%F A211603 T(n,k) = A111492(n,n-k). - _R. J. Mathar_, Mar 07 2013
%e A211603 T(3,1) = 3 because we have (1)(2,3), (2)(1,3), (3)(1,2).
%e A211603 1;
%e A211603 2, 3;
%e A211603 6, 8, 6;
%e A211603 24, 30, 20, 10;
%e A211603 120, 144, 90, 40, 15;
%e A211603 720, 840, 504, 210, 70, 21;
%e A211603 5040, 5760, 3360, 1344, 420, 112, 28;
%e A211603 40320, 45360, 25920, 10080, 3024, 756, 168, 36;
%e A211603 362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45;
%p A211603 T:= (n, k)-> binomial(n, k)*(n-k-1)!:
%p A211603 seq(seq(T(n,k), k=0..n-2), n=2..12);  # _Alois P. Heinz_, Feb 10 2013
%t A211603 nn=10;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[ Series[Exp[y x](Log[1/(1-x)]-x),{x,0,nn}],{x,y}]]//Grid
%Y A211603 Cf. A006231 (row sums), A008290, A092271, A111492, A238363.
%K A211603 nonn,tabl,easy
%O A211603 2,2
%A A211603 _Geoffrey Critzer_, Feb 10 2013