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A208956 Triangular array read by rows. T(n,k) is the number of n-permutations that have at least k fixed points with n >= 1 and 1 <= k <= n.

%I A208956 #18 Dec 14 2021 18:48:53
%S A208956 1,1,1,4,1,1,15,7,1,1,76,31,11,1,1,455,191,56,16,1,1,3186,1331,407,92,
%T A208956 22,1,1,25487,10655,3235,771,141,29,1,1,229384,95887,29143,6883,1339,
%U A208956 205,37,1,1,2293839,958879,291394,68914,13264,2176,286,46,1,1
%N A208956 Triangular array read by rows. T(n,k) is the number of n-permutations that have at least k fixed points with n >= 1 and 1 <= k <= n.
%C A208956 Row sums = n!
%H A208956 Alois P. Heinz, <a href="/A208956/b208956.txt">Rows n = 1..141, flattened</a>
%F A208956 E.g.f. for column k: 1/(1-x) - D(x)*Sum_{i=0..k-1} x^i/i! where D(x) is the e.g.f. for A000166.
%F A208956 T(n,k) = Sum_{i=k..n} C(n,i)*A000166(n-i). - _Alois P. Heinz_, Apr 22 2013
%e A208956 Triangle begins:
%e A208956      1;
%e A208956      1,    1;
%e A208956      4,    1,   1;
%e A208956     15,    7,   1,  1;
%e A208956     76,   31,  11,  1,  1;
%e A208956    455,  191,  56, 16,  1, 1;
%e A208956   3186, 1331, 407, 92, 22, 1, 1;
%e A208956   ...
%p A208956 b:= proc(n) b(n):= `if`(n<2, 1-n, (n-1)*(b(n-1)+b(n-2))) end:
%p A208956 T:= (n, k)-> add(binomial(n, i)*b(n-i), i=k..n):
%p A208956 seq(seq(T(n,k), k=1..n), n=1..12);  # _Alois P. Heinz_, Apr 22 2013
%t A208956 f[list_] := Select[list,#>0&]; Map[f,Transpose[Table[nn=10; d=Exp[-x]/(1-x); p=1/(1-x); s=Sum[x^i/i!,{i,0,n}]; Drop[Range[0,nn]! CoefficientList[Series[p-s d, {x,0,nn}], x], 1], {n,0,9}]]]//Flatten
%Y A208956 Cf. A002467 (column 1), A155521 (column 2).
%K A208956 nonn,tabl
%O A208956 1,4
%A A208956 _Geoffrey Critzer_, Mar 03 2012