A365674 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.

1, 1, 1, 1, 4, 4, 1, 10, 34, 34, 1, 20, 154, 496, 496, 1, 35, 504, 3520, 11056, 11056, 1, 56, 1344, 16960, 112816, 349504, 349504, 1, 84, 3108, 63580, 748616, 4841200, 14873104, 14873104, 1, 120, 6468, 199408, 3739736, 42238560, 268304464, 819786496, 819786496

Offset: 0

Author

Peter Luschny, Sep 30 2023

Keywords

• nonn
• tabl

Comments

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.

Examples

			[0] 1;
[1] 1,  1;
[2] 1,  4,    4;
[3] 1, 10,   34,    34;
[4] 1, 20,  154,   496,    496;
[5] 1, 35,  504,  3520,  11056,   11056;
[6] 1, 56, 1344, 16960, 112816,  349504,   349504;
[7] 1, 84, 3108, 63580, 748616, 4841200, 14873104, 14873104;
		

Crossrefs

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

Programs

Maple

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