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A126885 T(n,k) = n*T(n,k-1) + k, with T(n,1) = 1, square array read by ascending antidiagonals (n >= 0, k >= 1).

%I A126885 #15 Mar 31 2025 12:39:48
%S A126885 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 A126885 38,112,179,120,28,8,1,9,51,194,453,543,247,36,9,1,10,66,310,975,1818,
%U A126885 1636,502,45,10,1,11,83,466,1865,4881,7279,4916,1013,55,11
%N A126885 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 A126885 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 A126885 T(n,k) = third entry in the vector M^k * (1, 0, 0), where M is the following 3 X 3 matrix:
%F A126885   1, 0, 0
%F A126885   1, 1, 0
%F A126885   1, 1, n.
%e A126885 Square array begins:
%e A126885   n\k | 1   2   3   4    5     6      7       8 ...
%e A126885   -------------------------------------------------
%e A126885     0 | 1   2   3   4    5     6      7       8 ... A000027
%e A126885     1 | 1   3   6  10   15    21     28      36 ... A000217
%e A126885     2 | 1   4  11  26   57   120    247     502 ... A000295
%e A126885     3 | 1   5  18  58  179   543   1636    4916 ... A000340
%e A126885     4 | 1   6  27 112  453  1818   7279   29124 ... A014825
%e A126885     5 | 1   7  38 194  975  4881  24412  122068 ... A014827
%e A126885     6 | 1   8  51 310 1865 11196  67183  403106 ... A014829
%e A126885     7 | 1   9  66 466 3267 22875 160132 1120932 ... A014830
%e A126885     8 | 1  10  83 668 5349 42798 342391 2739136 ... A014831
%e A126885     ...
%o A126885 (Maxima)
%o A126885 T(n, k) := if k = 1 then 1 else n*T(n, k - 1) + k$
%o A126885 create_list(T(n - k + 1, k), n, 0, 20, k, 1, n + 1);
%o A126885 /* _Franck Maminirina Ramaharo_, Jan 26 2019 */
%Y A126885 Antidiagonal sums are A134195.
%Y A126885 Main diagonal gives A062805.
%Y A126885 Cf. A000027, A000217, A000295, A000340, A014825, A014827, A014829, A014830, A014831.
%K A126885 nonn,easy,tabl
%O A126885 0,3
%A A126885 _Gary W. Adamson_, Dec 30 2006
%E A126885 Edited and name clarified by _Franck Maminirina Ramaharo_, Jan 26 2019