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A228335 Triangle read by rows: the Y-transformation of the Catalan triangle A033184.

%I A228335 #55 Mar 02 2020 01:03:06
%S A228335 1,1,1,3,6,1,14,40,15,1,84,300,175,28,1,594,2475,1925,504,45,1,4719,
%T A228335 22022,21021,7644,1155,66,1,40898,208208,231868,107016,23100,2288,91,
%U A228335 1,379236,2068560,2598960,1439424,403920,58344,4095,120,1,3711916,21414900,29651400,18976896,6523308,1247103,129675,6800,153,1
%N A228335 Triangle read by rows: the Y-transformation of the Catalan triangle A033184.
%C A228335 From _Roger Ford_, Feb 08 2020: (Start)
%C A228335 For 2n-step octant walks, the numbers for the x axis ending locations are equal to numbers in row n of the example triangle.
%C A228335 Example: For n = 2, all 4-step octant walks starting at (0,0) and ending on the x axis are as follows:
%C A228335   EWEW EEWW   EEEW EWEE EEWE   EEEE
%C A228335   ENSW        EENS ENSE ENES
%C A228335   (0,0)       (2,0)            (4,0) - x axis ending location
%C A228335   3           6                1     - number of walks
%C A228335   These numbers match row 2 of the example triangle. (End)
%H A228335 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 A228335 Yidong Sun and Fei Ma, <a href="https://doi.org/10.37236/3701">Some new binomial sums related to the Catalan triangle</a>, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.
%e A228335 Triangle begins:
%e A228335     1;
%e A228335     1,    1;
%e A228335     3,    6,    1;
%e A228335    14,   40,   15,    1;
%e A228335    84,  300,  175,   28,    1;
%e A228335   594, 2475, 1925,  504,   45,    1;
%e A228335   ...
%t A228335 nn = 9;
%t A228335 c[n_, k_] := Binomial[2n-k, n] (k+1)/(n+1);
%t A228335 a[0, 0] = 1;
%t A228335 a[n_, k_] := Table[c[n+k+i, 2k+j], {i, 2}, {j, 2}] // Det;
%t A228335 Table[a[n, k], {n, 0, nn}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 12 2018 *)
%o A228335 (PARI) C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
%o A228335 T(n, k) = matdet(matrix(2, 2, i, j, C(n+k+i, 2*k+j))); \\ _Michel Marcus_, Feb 13 2014
%Y A228335 Cf. A033184, A228334, A228336, A228337.
%K A228335 nonn,tabl
%O A228335 0,4
%A A228335 _N. J. A. Sloane_, Aug 26 2013
%E A228335 More terms from _Michel Marcus_, Feb 13 2014
%E A228335 A-number for Catalan triangle changed by _Michel Marcus_, Feb 13 2014