A366138 Triangle read by rows. T(n, k) = A000326(n - k + 1) * T(n, k - 1) + T(n - 1, k) for 0 < k < n. T(n, 0) = 1 and T(n, n) = T(n, n - 1) if n > 0.

1, 1, 1, 1, 6, 6, 1, 18, 96, 96, 1, 40, 576, 2976, 2976, 1, 75, 2226, 29688, 151416, 151416, 1, 126, 6636, 175680, 2259576, 11449296, 11449296, 1, 196, 16632, 757800, 18931176, 238623408, 1204566336, 1204566336

Offset: 0

Author

Peter Luschny, Oct 01 2023

Keywords

• nonn
• tabl

Comments

This a weighted generalized Catalan triangle (A365673) with the pentagonal numbers as weights.

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 1,   1;
[2] 1,   6,     6;
[3] 1,  18,    96,     96;
[4] 1,  40,   576,   2976,     2976;
[5] 1,  75,  2226,  29688,   151416,    151416;
[6] 1, 126,  6636, 175680,  2259576,  11449296,   11449296;
[7] 1, 196, 16632, 757800, 18931176, 238623408, 1204566336, 1204566336;
		

Crossrefs

Cf. A000326, A126151 (main diagonal), A365673.

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)*(3*n - 3*k + 2))/2) * T(n, k - 1) + T(n - 1, k) fi fi end:
seq(seq(T(n, k), k = 0..n), n = 0..8);