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.
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, 26, 177, 1000, 2329, 720, 1, 1, 1, 37, 326, 2913, 14968, 27949, 5040, 1, 1, 1, 50, 541, 6776, 58017, 269488, 391285, 40320, 1, 1, 1, 65, 834, 13609, 168376, 1393137, 5659120, 6260561, 362880, 1
Offset: 0
Examples
Row n=0: 1, 1, 1, 1, 1, 1, 1, 1, ... Row n=1: 1, 1, 1, 1, 1, 1, 1, 1, ... Row n=2: 1, 2, 5, 10, 17, 26, 37, 50, ... Row n=3: 1, 6, 29, 82, 177, 326, 541, 834, ... Row n=4: 1, 24, 233, 1000, 2913, 6776, 13609, 24648, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Programs
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Maple
T:= (n, k)-> `if`(n=0, 1, LinearAlgebra[Permanent]( Matrix(n, (i, j)-> `if`(i=j, 1, k)))): seq(seq(T(n, d-n), n=0..d), d=0..10); # Alois P. Heinz, Jul 09 2017 # second Maple program: b:= proc(n, k) b(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)* (b(n-1, 0)+b(n-2, 0))), binomial(n, k)*b(n-k, 0)) end: T:= proc(n, k) T(n, k):= add(b(n, j)*k^(n-j), j=0..n) end: seq(seq(T(n, d-n), n=0..d), d=0..10); # Alois P. Heinz, Jul 09 2017 -
Mathematica
T[0, _] = 1; T[n_, k_] := Permanent[Table[If[i == j, 1, k], {i, n}, {j, n}]]; Table[T[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 07 2019 *) -
PARI
T(n,k) = matpermanent(matrix(n, n, i, j, if (i==j, 1, k))); matrix(10, 10, n, k, T(n,k)) \\ Michel Marcus, Dec 07 2019
Formula
T(n, k) = Sum_{j=0..n} A008290(n, j)*k^(n-j).
Extensions
3 terms corrected and more terms from Alois P. Heinz, Jul 09 2017