%I #24 Feb 09 2021 08:19:34
%S 1,0,0,0,0,1,0,0,2,0,1,2,3,2,1,4,12,14,8,6,0,29,68,82,54,25,6,1,206,
%T 496,546,376,170,48,12,0,1708,3960,4349,2922,1353,430,98,12,1,15702,
%U 35816,38632,26048,12084,4052,982,160,20,0,159737,358786,383523,257552,120919,41508,10647,1998,270,20,1
%N Number T(n,k) of permutations p of [n] with no fixed points such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.
%H Alois P. Heinz, <a href="/A323671/b323671.txt">Rows n = 0..23, flattened</a>
%F Sum_{k=1..n} T(n,k) = A296050(n).
%e T(4,0) = 1: 3412.
%e T(4,1) = 2: 3421, 4312.
%e T(4,2) = 3: 2413, 3142, 4321.
%e T(4,3) = 2: 2341, 4123.
%e T(4,4) = 1: 2143.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 0;
%e 0, 0, 1;
%e 0, 0, 2, 0;
%e 1, 2, 3, 2, 1;
%e 4, 12, 14, 8, 6, 0;
%e 29, 68, 82, 54, 25, 6, 1;
%e 206, 496, 546, 376, 170, 48, 12, 0;
%e 1708, 3960, 4349, 2922, 1353, 430, 98, 12, 1;
%e 15702, 35816, 38632, 26048, 12084, 4052, 982, 160, 20, 0;
%e ...
%p b:= proc(s) option remember; expand((n-> `if`(n=0, 1, add(
%p (t-> `if`(t=0, 0, `if`(t=1, x, 1)*b(s minus {j}))
%p )(abs(n-j)), j=s)))(nops(s)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b({$1..n})):
%p seq(T(n), n=0..12);
%t b[s_] := b[s] = Expand[Function[n, If[n==0, 1, Sum[Function[t, If[t==0, 0, If[t==1, x, 1]*b[s~Complement~{j}]]][Abs[n-j]], {j, s}]]][Length[s]]];
%t T[n_] := PadRight[CoefficientList[b[Range[n]], x], n+1];
%t T /@ Range[0, 12] // Flatten (* _Jean-François Alcover_, Feb 09 2021, after _Alois P. Heinz_ *)
%Y Column k=0 gives A001883.
%Y Row sums give A000166.
%Y Main diagonal and lower diagonal give A059841, A110660.
%Y Cf. A296050, A320582.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Jan 23 2019