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A306234 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.

%I A306234 #58 Feb 15 2021 04:41:39
%S A306234 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 A306234 411,455,411,179,49,9,1,1,11,76,375,1306,2921,3186,2921,1306,375,76,
%U A306234 11,1,1,13,109,679,3181,10757,23633,25487,23633,10757,3181,679,109,13,1
%N A306234 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 A306234 Alois P. Heinz, <a href="/A306234/b306234.txt">Rows n = 1..142, flattened</a>
%H A306234 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A306234 T(n,k) = T(n,-k).
%F A306234 T(n,k) = -1/|k|! * Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!.
%F A306234 T(n,k) = (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1).
%F A306234 T(n+1,n) = 1.
%F A306234 T(n,k) = A306461(n,k) / |k|!.
%F A306234 Sum_{k=1-n..n-1} |k|! * T(n,k) = A306455(n).
%e A306234 Triangle T(n,k) begins:
%e A306234   :                                 1                              ;
%e A306234   :                           1,    1,    1                        ;
%e A306234   :                     1,    3,    4,    3,    1                  ;
%e A306234   :               1,    5,   13,   15,   13,    5,   1             ;
%e A306234   :          1,   7,   28,   67,   76,   67,   28,   7,  1         ;
%e A306234   :      1,  9,  49,  179,  411,  455,  411,  179,  49,  9,  1     ;
%e A306234   :  1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1  ;
%p A306234 b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),
%p A306234       add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))
%p A306234     end:
%p A306234 T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, {})):
%p A306234 seq(T(n), n=1..8);
%p A306234 # second Maple program:
%p A306234 T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n)/abs(k)!:
%p A306234 seq(seq(T(n, k), k=1-n..n-1), n=1..9);
%t A306234 T[n_, k_] := (-1/Abs[k]!) Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}];
%t A306234 Table[T[n, k], {n, 1, 9}, {k, 1-n, n-1}] // Flatten (* _Jean-François Alcover_, Feb 15 2021 *)
%Y A306234 Columns k=0-10 give (offsets may differ): A002467, A180191, A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.
%Y A306234 Row sums give A306525.
%Y A306234 T(n+1,n) gives A000012.
%Y A306234 T(n+2,n) gives A005408.
%Y A306234 T(n+2,n-1) gives A056107.
%Y A306234 T(2n,n) gives A324361.
%Y A306234 Cf. A000142, A306455, A306461, A324224, A324362.
%K A306234 nonn,tabf
%O A306234 1,6
%A A306234 _Alois P. Heinz_, Feb 17 2019