%I #7 Sep 30 2023 20:58:06

%S 1,1,1,1,4,4,1,10,34,34,1,20,154,496,496,1,35,504,3520,11056,11056,1,

%T 56,1344,16960,112816,349504,349504,1,84,3108,63580,748616,4841200,

%U 14873104,14873104,1,120,6468,199408,3739736,42238560,268304464,819786496,819786496

%N Triangle read by rows. T(n, k) = ((n - k + 1)*(n - k + 2)/2) * T(n, k - 1) + T(n - 1, k) for 0 < k < n, T(n, 0) = 1 and T(n, n) = T(n, n - 1) for n > 0.

%C This triangle is associated to the case n = 3 of A365673 and has as weight function the triangular numbers A000217. The numbers on its main diagonal are the reduced tangent numbers A002105. For details see A365673.

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 4, 4;

%e [3] 1, 10, 34, 34;

%e [4] 1, 20, 154, 496, 496;

%e [5] 1, 35, 504, 3520, 11056, 11056;

%e [6] 1, 56, 1344, 16960, 112816, 349504, 349504;

%e [7] 1, 84, 3108, 63580, 748616, 4841200, 14873104, 14873104;

%p T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k - 1) else ((n - k + 1)*(n - k + 2)/2) * T( n, k - 1) + T( n - 1, k) fi fi end:

%p seq(print(seq(T(n, k), k = 0..n)), n = 0..8);

%Y Cf. A002105 (main diagonal), A365673 (case n=3), A000217 (weight).

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Sep 30 2023