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