A365672 Triangle read by rows. T(n, k) = 1 if k = 0, equals T(n, k-1) if k = n, and otherwise is (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k).

1, 1, 1, 1, 7, 7, 1, 22, 139, 139, 1, 50, 889, 5473, 5473, 1, 95, 3549, 58708, 357721, 357721, 1, 161, 10794, 360940, 5771821, 34988647, 34988647, 1, 252, 27426, 1595110, 50434901, 791512162, 4784061619, 4784061619

Offset

0,5

Comments

This triangle is described by Peter Bala (see link).

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

Links

Table of n, a(n) for n=0..35.

Peter Bala, A triangle for calculating A126156.

Example

Triangle T(n, k) starts:
[0] 1;
[1] 1,   1;
[2] 1,   7,     7;
[3] 1,  22,   139,     139;
[4] 1,  50,   889,    5473,     5473;
[5] 1,  95,  3549,   58708,   357721,    357721;
[6] 1, 161, 10794,  360940,  5771821,  34988647,   34988647;
[7] 1, 252, 27426, 1595110, 50434901, 791512162, 4784061619, 4784061619;

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) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k) fi fi end:

Crossrefs

Cf. A000384, A126156 (main diagonal), A365673 (general case).

Sequence in context: A093781 A232649 A232650 * A108390 A102400 A144860

Adjacent sequences: A365669 A365670 A365671 * A365673 A365674 A365675

Keyword

nonn,tabl

Author

Peter Luschny, Sep 29 2023

Status

approved