A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 9, 1, 4, 10, 16, 24, 44, 1, 5, 17, 38, 65, 120, 265, 1, 6, 26, 78, 168, 326, 720, 1854, 1, 7, 37, 142, 393, 872, 1957, 5040, 14833, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 133496, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1334961

Offset: 0

Author

Philippe Deléham, Dec 12 2003

Keywords

• easy
• nonn
• tabl

Comments

Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - Max Alekseyev, Mar 06 2018

Examples

			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, ...
...
		

Crossrefs

Columns: A000166, A000142, A000522, A010842, A053486, A053487, A080954.

Main diagonal gives A217701.

Cf. A080955, A008290, A292977.

Programs

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 *)

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

Extensions

Edited and changed offset for k to -1 by Max Alekseyev, Mar 08 2018