A126885 - OEIS

%I #13 Jan 27 2019 05:11:23

%S 1,1,2,1,3,3,1,4,6,4,1,5,11,10,5,1,6,18,26,15,6,1,7,27,58,57,21,7,1,8,

%T 38,112,179,120,28,8,1,9,51,194,453,543,247,36,9,1,10,66,310,975,1818,

%U 1636,502,45,10,1,11,83,466,1865,4881,7279,4916,1013,55,11

%N T(n,k) = n*T(n,k-1) + k, with T(n,1) = 1, square array read by ascending antidiagonals (n >= 0, k >= 1).

%F T(1,k) = k*(k + 1)/2, and T(n,k) = (k - (k + 1)*n + n^(k + 1))/(n^2 - 2*n + 1) elsewhere.

%F T(n,k) = third entry in the vector M^k * (1, 0, 0), where M is the following 3 X 3 matrix:

%F   1, 0, 0

%F   1, 1, 0

%F   1, 1, n.

%e Square array begins:

%e   n\k | 1   2   3   4    5     6      7       8 ...

%e   -------------------------------------------------

%e     0 | 1   2   3   4    5     6      7       8 ... A000027

%e     1 | 1   3   6  10   15    21     28      36 ... A000217

%e     2 | 1   4  11  26   57   120    247     502 ... A000295

%e     3 | 1   5  18  58  179   543   1636    4916 ... A000340

%e     4 | 1   6  27 112  453  1818   7279   29124 ... A014825

%e     5 | 1   7  38 194  975  4881  24412  122068 ... A014827

%e     6 | 1   8  51 310 1865 11196  67183  403106 ... A014829

%e     7 | 1   9  66 466 3267 22875 160132 1120932 ... A014830

%e     8 | 1  10  83 668 5349 42798 342391 2739136 ... A014831

%e     ...

%o (Maxima)

%o T(n, k) := if k = 1 then 1 else n*T(n, k - 1) + k$

%o create_list(T(n - k + 1, k), n, 0, 20, k, 1, n + 1);

%o /* _Franck Maminirina Ramaharo_, Jan 26 2019 */

%Y Antidiagonal sums: A134195. - _Gary W. Adamson_, Oct 12 2007

%Y Cf. A000027, A000217, A000295, A000340, A014825, A014827, A014829, A014830, A014831.

%K nonn,easy,tabl

%O 0,3

%A _Gary W. Adamson_, Dec 30 2006

%E Edited and name clarified by _Franck Maminirina Ramaharo_, Jan 26 2019