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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.

%I A366138 #5 Oct 01 2023 07:31:54
%S A366138 1,1,1,1,6,6,1,18,96,96,1,40,576,2976,2976,1,75,2226,29688,151416,
%T A366138 151416,1,126,6636,175680,2259576,11449296,11449296,1,196,16632,
%U A366138 757800,18931176,238623408,1204566336,1204566336
%N 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.
%C A366138 This a weighted generalized Catalan triangle (A365673) with the pentagonal numbers as weights.
%e A366138 Triangle T(n, k) starts:
%e A366138 [0] 1;
%e A366138 [1] 1,   1;
%e A366138 [2] 1,   6,     6;
%e A366138 [3] 1,  18,    96,     96;
%e A366138 [4] 1,  40,   576,   2976,     2976;
%e A366138 [5] 1,  75,  2226,  29688,   151416,    151416;
%e A366138 [6] 1, 126,  6636, 175680,  2259576,  11449296,   11449296;
%e A366138 [7] 1, 196, 16632, 757800, 18931176, 238623408, 1204566336, 1204566336;
%p A366138 T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
%p A366138 else (((n - k + 1)*(3*n - 3*k + 2))/2) * T(n, k - 1) + T(n - 1, k) fi fi end:
%p A366138 seq(seq(T(n, k), k = 0..n), n = 0..8);
%Y A366138 Cf. A000326, A126151 (main diagonal), A365673.
%K A366138 nonn,tabl
%O A366138 0,5
%A A366138 _Peter Luschny_, Oct 01 2023