OFFSET
0,8
COMMENTS
Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - Max Alekseyev, Mar 06 2018
FORMULA
For n > 0, k >= -1, T(n,k) is the permanent of the n X n matrix with k+1 on the diagonal and 1 elsewhere.
T(0,k) = 1.
T(n,k) = Sum_{j>=0} A008290(n,j) * (k+1)^j.
T(n,k) = n*T(n-1, k) + k^n .
T(n,k) = n! * Sum_{j=0..n} k^j/j!.
E.g.f. for k-th column: exp(k*x)/(1-x).
Assuming n >= 0, k >= 0: T(n, k) = exp(k-1)*Gamma(n+1, k-1). - Peter Luschny, Dec 24 2021
EXAMPLE
n\k -1 0 1 2 3 4 5 6 ...
----------------------------------------------
0 | 1, 1, 1, 1, 1, 1, 1, 1, ...
1 | 0, 1, 2, 3, 4, 5, 6, 7, ...
2 | 1, 2, 5, 10, 17, 26, 37, 50, ...
3 | 2, 6, 16, 38, 78, 152, 236, 366, ...
4 | 9, 24, 65, 168, 393, 824, 1569, 2760, ...
...
MATHEMATICA
(* Assuming offset (0, 0): *)
T[n_, k_] := Exp[k - 1] Gamma[n + 1, k - 1];
Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 24 2021 *)
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Dec 12 2003
EXTENSIONS
Edited and changed offset for k to -1 by Max Alekseyev, Mar 08 2018
STATUS
approved