%I #58 Feb 15 2021 04:41:39

%S 1,1,1,1,1,3,4,3,1,1,5,13,15,13,5,1,1,7,28,67,76,67,28,7,1,1,9,49,179,

%T 411,455,411,179,49,9,1,1,11,76,375,1306,2921,3186,2921,1306,375,76,

%U 11,1,1,13,109,679,3181,10757,23633,25487,23633,10757,3181,679,109,13,1

%N Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n] divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

%H Alois P. Heinz, <a href="/A306234/b306234.txt">Rows n = 1..142, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%F T(n,k) = T(n,-k).

%F T(n,k) = -1/|k|! * Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!.

%F T(n,k) = (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1).

%F T(n+1,n) = 1.

%F T(n,k) = A306461(n,k) / |k|!.

%F Sum_{k=1-n..n-1} |k|! * T(n,k) = A306455(n).

%e Triangle T(n,k) begins:

%e : 1 ;

%e : 1, 1, 1 ;

%e : 1, 3, 4, 3, 1 ;

%e : 1, 5, 13, 15, 13, 5, 1 ;

%e : 1, 7, 28, 67, 76, 67, 28, 7, 1 ;

%e : 1, 9, 49, 179, 411, 455, 411, 179, 49, 9, 1 ;

%e : 1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1 ;

%p b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),

%p add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, {})):

%p seq(T(n), n=1..8);

%p # second Maple program:

%p T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n)/abs(k)!:

%p seq(seq(T(n, k), k=1-n..n-1), n=1..9);

%t T[n_, k_] := (-1/Abs[k]!) Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}];

%t Table[T[n, k], {n, 1, 9}, {k, 1-n, n-1}] // Flatten (* _Jean-François Alcover_, Feb 15 2021 *)

%Y Columns k=0-10 give (offsets may differ): A002467, A180191, A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.

%Y Row sums give A306525.

%Y T(n+1,n) gives A000012.

%Y T(n+2,n) gives A005408.

%Y T(n+2,n-1) gives A056107.

%Y T(2n,n) gives A324361.

%Y Cf. A000142, A306455, A306461, A324224, A324362.

%K nonn,tabf

%O 1,6

%A _Alois P. Heinz_, Feb 17 2019