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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.
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%I #28 Dec 07 2019 17:24:09

%S 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 26,177,1000,2329,720,1,1,1,37,326,2913,14968,27949,5040,1,1,1,50,541,

%U 6776,58017,269488,391285,40320,1,1,1,65,834,13609,168376,1393137,5659120,6260561,362880,1

%N 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 Alois P. Heinz, <a href="/A090628/b090628.txt">Antidiagonals n = 0..140, flattened</a>

%F T(n, k) = Sum_{j=0..n} A008290(n, j)*k^(n-j).

%e Row n=0: 1, 1, 1, 1, 1, 1, 1, 1, ...

%e Row n=1: 1, 1, 1, 1, 1, 1, 1, 1, ...

%e Row n=2: 1, 2, 5, 10, 17, 26, 37, 50, ...

%e Row n=3: 1, 6, 29, 82, 177, 326, 541, 834, ...

%e Row n=4: 1, 24, 233, 1000, 2913, 6776, 13609, 24648, ...

%p T:= (n, k)-> `if`(n=0, 1, LinearAlgebra[Permanent](

%p Matrix(n, (i, j)-> `if`(i=j, 1, k)))):

%p seq(seq(T(n, d-n), n=0..d), d=0..10); # _Alois P. Heinz_, Jul 09 2017

%p # second Maple program:

%p b:= proc(n, k) b(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*

%p (b(n-1, 0)+b(n-2, 0))), binomial(n, k)*b(n-k, 0))

%p end:

%p T:= proc(n, k) T(n, k):= add(b(n, j)*k^(n-j), j=0..n) end:

%p seq(seq(T(n, d-n), n=0..d), d=0..10); # _Alois P. Heinz_, Jul 09 2017

%t T[0, _] = 1;

%t T[n_, k_] := Permanent[Table[If[i == j, 1, k], {i, n}, {j, n}]];

%t Table[T[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 07 2019 *)

%o (PARI) T(n,k) = matpermanent(matrix(n, n, i, j, if (i==j, 1, k)));

%o matrix(10, 10, n, k, T(n,k)) \\ _Michel Marcus_, Dec 07 2019

%Y Cf. A008290.

%Y Columns: A000012, A000142, A000354.

%K easy,nonn,tabl

%O 0,9

%A _Philippe Deléham_, Dec 13 2003

%E 3 terms corrected and more terms from _Alois P. Heinz_, Jul 09 2017