%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