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

%I A089258 #26 Dec 24 2021 13:09:37
%S A089258 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,
%T A089258 6,26,78,168,326,720,1854,1,7,37,142,393,872,1957,5040,14833,1,8,50,
%U A089258 236,824,2208,5296,13700,40320,133496,1,9,65,366,1569,5144,13977,37200,109601,362880,1334961
%N A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.
%C A089258 Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - _Max Alekseyev_, Mar 06 2018
%F A089258 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.
%F A089258 T(0,k) = 1.
%F A089258 T(n,k) = Sum_{j>=0} A008290(n,j) * (k+1)^j.
%F A089258 T(n,k) = n*T(n-1, k) + k^n .
%F A089258 T(n,k) = n! * Sum_{j=0..n} k^j/j!.
%F A089258 E.g.f. for k-th column: exp(k*x)/(1-x).
%F A089258 Assuming n >= 0, k >= 0: T(n, k) = exp(k-1)*Gamma(n+1, k-1). - _Peter Luschny_, Dec 24 2021
%e A089258 n\k -1   0   1    2    3    4     5     6  ...
%e A089258 ----------------------------------------------
%e A089258 0  | 1,  1,  1,   1,   1,   1,    1,    1, ...
%e A089258 1  | 0,  1,  2,   3,   4,   5,    6,    7, ...
%e A089258 2  | 1,  2,  5,  10,  17,  26,   37,   50, ...
%e A089258 3  | 2,  6, 16,  38,  78, 152,  236,  366, ...
%e A089258 4  | 9, 24, 65, 168, 393, 824, 1569, 2760, ...
%e A089258 ...
%t A089258 (* Assuming offset (0, 0): *)
%t A089258 T[n_, k_] := Exp[k - 1] Gamma[n + 1, k - 1];
%t A089258 Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten  (* _Peter Luschny_, Dec 24 2021 *)
%Y A089258 Columns: A000166, A000142, A000522, A010842, A053486, A053487, A080954.
%Y A089258 Main diagonal gives A217701.
%Y A089258 Cf. A080955, A008290, A292977.
%K A089258 easy,nonn,tabl
%O A089258 0,8
%A A089258 _Philippe Deléham_, Dec 12 2003
%E A089258 Edited and changed offset for k to -1 by _Max Alekseyev_, Mar 08 2018