%I #29 Nov 16 2023 15:07:50

%S 1,1,2,1,1,4,6,4,1,1,6,18,24,18,6,1,1,8,36,96,120,96,36,8,1,1,10,60,

%T 240,600,720,600,240,60,10,1,1,12,90,480,1800,4320,5040,4320,1800,480,

%U 90,12,1,1,14,126,840,4200,15120,35280,40320,35280,15120,4200,840,126,14,1

%N Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

%H Alois P. Heinz, <a href="/A324224/b324224.txt">Rows n = 1..100, flattened</a>

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

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

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

%F T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.

%F T(n,k) = 1/|k|! * A324225(n,k).

%F E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.

%F Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).

%e Triangle T(n,k) begins:

%e : 1 ;

%e : 1, 2, 1 ;

%e : 1, 4, 6, 4, 1 ;

%e : 1, 6, 18, 24, 18, 6, 1 ;

%e : 1, 8, 36, 96, 120, 96, 36, 8, 1 ;

%e : 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1 ;

%e : 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1 ;

%p b:= proc(s, c) option remember; (n-> `if`(n=0, c,

%p add(b(s minus {i}, c+x^(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}, 0)):

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

%p # second Maple program:

%p egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):

%p T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):

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

%p # third Maple program:

%p T:= (n, k)-> (t-> `if`(t<n, (n-t)*(n-1)!/t!, 0))(abs(k)):

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

%t T[n_, k_] := With[{t = Abs[k]}, If[t<n, (n-t)(n-1)!/t!, 0]];

%t Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Mar 25 2021, after 3rd Maple program *)

%Y Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.

%Y Row sums give A306495(n-1).

%Y Cf. A132159 (right part of triangle), A306234, A324225.

%K nonn,tabf

%O 1,3

%A _Alois P. Heinz_, Feb 18 2019