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A090628 Square array T(n,k) (row n, column k) read by antidiagonals defined by: T(n,k) is the permanent of the n X n matrix with 1 on the diagonal and k elsewhere; T(0,k)=1.

%I A090628 #28 Dec 07 2019 17:24:09
%S A090628 1,1,1,1,1,1,1,1,2,1,1,1,5,6,1,1,1,10,29,24,1,1,1,17,82,233,120,1,1,1,
%T A090628 26,177,1000,2329,720,1,1,1,37,326,2913,14968,27949,5040,1,1,1,50,541,
%U A090628 6776,58017,269488,391285,40320,1,1,1,65,834,13609,168376,1393137,5659120,6260561,362880,1
%N A090628 Square array T(n,k) (row n, column k) read by antidiagonals defined by: T(n,k) is the permanent of the n X n matrix with 1 on the diagonal and k elsewhere; T(0,k)=1.
%H A090628 Alois P. Heinz, <a href="/A090628/b090628.txt">Antidiagonals n = 0..140, flattened</a>
%F A090628 T(n, k) = Sum_{j=0..n} A008290(n, j)*k^(n-j).
%e A090628 Row n=0: 1,  1,   1,    1,    1,    1,     1,     1, ...
%e A090628 Row n=1: 1,  1,   1,    1,    1,    1,     1,     1, ...
%e A090628 Row n=2: 1,  2,   5,   10,   17,   26,    37,    50, ...
%e A090628 Row n=3: 1,  6,  29,   82,  177,  326,   541,   834, ...
%e A090628 Row n=4: 1, 24, 233, 1000, 2913, 6776, 13609, 24648, ...
%p A090628 T:= (n, k)-> `if`(n=0, 1, LinearAlgebra[Permanent](
%p A090628               Matrix(n, (i, j)-> `if`(i=j, 1, k)))):
%p A090628 seq(seq(T(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Jul 09 2017
%p A090628 # second Maple program:
%p A090628 b:= proc(n, k) b(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
%p A090628       (b(n-1, 0)+b(n-2, 0))), binomial(n, k)*b(n-k, 0))
%p A090628     end:
%p A090628 T:= proc(n, k) T(n, k):= add(b(n, j)*k^(n-j), j=0..n) end:
%p A090628 seq(seq(T(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Jul 09 2017
%t A090628 T[0, _] = 1;
%t A090628 T[n_, k_] := Permanent[Table[If[i == j, 1, k], {i, n}, {j, n}]];
%t A090628 Table[T[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 07 2019 *)
%o A090628 (PARI) T(n,k) = matpermanent(matrix(n, n, i, j, if (i==j, 1, k)));
%o A090628 matrix(10, 10, n, k, T(n,k)) \\ _Michel Marcus_, Dec 07 2019
%Y A090628 Cf. A008290.
%Y A090628 Columns: A000012, A000142, A000354.
%K A090628 easy,nonn,tabl
%O A090628 0,9
%A A090628 _Philippe Deléham_, Dec 13 2003
%E A090628 3 terms corrected and more terms from _Alois P. Heinz_, Jul 09 2017