A365672 - OEIS

%I #9 Oct 01 2023 07:13:10

%S 1,1,1,1,7,7,1,22,139,139,1,50,889,5473,5473,1,95,3549,58708,357721,

%T 357721,1,161,10794,360940,5771821,34988647,34988647,1,252,27426,

%U 1595110,50434901,791512162,4784061619,4784061619

%N 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).

%C This triangle is described by _Peter Bala_ (see link).

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

%H Peter Bala, <a href="/A126156/a126156.pdf">A triangle for calculating A126156</a>.

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 1,   1;

%e [2] 1,   7,     7;

%e [3] 1,  22,   139,     139;

%e [4] 1,  50,   889,    5473,     5473;

%e [5] 1,  95,  3549,   58708,   357721,    357721;

%e [6] 1, 161, 10794,  360940,  5771821,  34988647,   34988647;

%e [7] 1, 252, 27426, 1595110, 50434901, 791512162, 4784061619, 4784061619;

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

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

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Sep 29 2023