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A324224 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.

%I A324224 #29 Nov 16 2023 15:07:50
%S A324224 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 A324224 240,600,720,600,240,60,10,1,1,12,90,480,1800,4320,5040,4320,1800,480,
%U A324224 90,12,1,1,14,126,840,4200,15120,35280,40320,35280,15120,4200,840,126,14,1
%N A324224 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 A324224 Alois P. Heinz, <a href="/A324224/b324224.txt">Rows n = 1..100, flattened</a>
%H A324224 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%H A324224 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_matrix">Permutation matrix</a>
%F A324224 T(n,k) = T(n,-k).
%F A324224 T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.
%F A324224 T(n,k) = 1/|k|! * A324225(n,k).
%F A324224 E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.
%F A324224 Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).
%e A324224 Triangle T(n,k) begins:
%e A324224   :                                 1                              ;
%e A324224   :                           1,    2,    1                        ;
%e A324224   :                     1,    4,    6,    4,    1                  ;
%e A324224   :               1,    6,   18,   24,   18,    6,   1             ;
%e A324224   :          1,   8,   36,   96,  120,   96,   36,   8,  1         ;
%e A324224   :      1, 10,  60,  240,  600,  720,  600,  240,  60, 10,  1     ;
%e A324224   :  1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1  ;
%p A324224 b:= proc(s, c) option remember; (n-> `if`(n=0, c,
%p A324224       add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
%p A324224     end:
%p A324224 T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):
%p A324224 seq(T(n), n=1..8);
%p A324224 # second Maple program:
%p A324224 egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):
%p A324224 T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
%p A324224 seq(seq(T(n, k), k=1-n..n-1), n=1..8);
%p A324224 # third Maple program:
%p A324224 T:= (n, k)-> (t-> `if`(t<n, (n-t)*(n-1)!/t!, 0))(abs(k)):
%p A324224 seq(seq(T(n, k), k=1-n..n-1), n=1..8);
%t A324224 T[n_, k_] := With[{t = Abs[k]}, If[t<n, (n-t)(n-1)!/t!, 0]];
%t A324224 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 A324224 Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.
%Y A324224 Row sums give A306495(n-1).
%Y A324224 Cf. A132159 (right part of triangle), A306234, A324225.
%K A324224 nonn,tabf
%O A324224 1,3
%A A324224 _Alois P. Heinz_, Feb 18 2019