%I #10 Jan 05 2024 13:46:32

%S 1,1,1,1,8,8,1,26,190,190,1,60,1270,9080,9080,1,115,5180,102320,

%T 725320,725320,1,196,15960,644960,12334600,87067520,87067520,1,308,

%U 40908,2894900,110761200,2080769120,14652451360,14652451360

%N Triangle read by rows. T(n, k) = A000566(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.

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

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 8, 8;

%e [3] 1, 26, 190, 190;

%e [4] 1, 60, 1270, 9080, 9080;

%e [5] 1, 115, 5180, 102320, 725320, 725320;

%e [6] 1, 196, 15960, 644960, 12334600, 87067520, 87067520;

%e [7] 1, 308, 40908, 2894900, 110761200, 2080769120, 14652451360, 14652451360;

%p T := proc(n, k) option remember;

%p if k = 0 then 1 else if k = n then T(n, k-1) else

%p (((5*k - 5*n - 2)*(k - n - 1))/2) * T(n, k - 1) + T(n - 1, k) fi fi end:

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

%t A366149[n_, k_] := A366149[n, k] = Which[k==0, 1, k==n, A366149[n, k-1], True, PolygonalNumber[7, n-k+1] A366149[n, k-1] + A366149[n-1, k]];

%t Table[A366149[n, k], {n,0,10}, {k,0,n}] (* _Paolo Xausa_, Jan 01 2024 *)

%Y Cf. A000566, A366150 (main diagonal), A365673 (general case).

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Oct 01 2023