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A228334 Triangle read by rows: the X-transformation of the Catalan triangle A033184.

%I A228334 #41 Aug 20 2018 08:07:14
%S A228334 1,0,1,0,3,1,0,14,10,1,0,84,90,21,1,0,594,825,308,36,1,0,4719,7865,
%T A228334 4004,780,55,1,0,40898,78078,49686,13650,1650,78,1,0,379236,804440,
%U A228334 606424,214200,37400,3094,105,1,0,3711916,8565960,7379904,3162816,724812,88179,5320,136,1
%N A228334 Triangle read by rows: the X-transformation of the Catalan triangle A033184.
%H A228334 Fangfang Cai, Qing-Hu Hou, Yidong Sun, Arthur L.B. Yang, <a href="https://arxiv.org/abs/1808.05736">Combinatorial identities related to 2×2 submatrices of recursive matrices</a>, arXiv:1808.05736 [math.CO], 2018.
%H A228334 Yidong Sun and Fei Ma, <a href="http://arxiv.org/abs/1305.2015">Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths</a>, arXiv preprint arXiv:1305.2015 [math.CO], 2013.
%H A228334 Yidong Sun and Fei Ma, <a href="http://arxiv.org/abs/1305.2017">Four transformations on the Catalan triangle</a>, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
%H A228334 Yidong Sun and Fei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p33">Some new binomial sums related to the Catalan triangle</a>, Electronic Journal of Combinatorics 21(1) (2014), #P1.33
%e A228334 Triangle begins:
%e A228334   1;
%e A228334   0,   1;
%e A228334   0,   3,   1;
%e A228334   0,  14,  10,   1;
%e A228334   0,  84,  90,  21,   1;
%e A228334   0, 594, 825, 308,  36,   1;
%e A228334   ...
%t A228334 nn = 9;
%t A228334 c[n_, k_] := Binomial[2n-k, n] (k+1)/(n+1);
%t A228334 a[0, 0] = 1;
%t A228334 a[n_, k_] := Table[c[n+k+i-1, 2k+j-1], {i, 1, 2}, {j, 1, 2}] // Det;
%t A228334 Table[a[n, k], {n, 0, nn}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 12 2018 *)
%o A228334 (PARI) C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
%o A228334 aX(nn) = {for (n = 0, nn, for (k = 0, n, print1(matdet(matrix(2, 2, i, j, C(n+k+i-1, 2*k+j-1))), ", ");); print(););} \\ _Michel Marcus_, Feb 13 2014
%Y A228334 Cf. A033184, A179898, A228335, A228336, A228337.
%K A228334 nonn,tabl
%O A228334 0,5
%A A228334 _N. J. A. Sloane_, Aug 26 2013
%E A228334 More terms from _Michel Marcus_, Feb 13 2014
%E A228334 A-number for Catalan triangle changed by _Michel Marcus_, Feb 13 2014