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A228337 Irregular triangle read by rows: the W-transformation of the Catalan triangle A033184.

%I A228337 #26 Aug 13 2018 09:08:03
%S A228337 1,2,4,1,10,4,20,21,1,56,70,6,140,238,50,1,420,792,210,8,1176,2604,
%T A228337 990,91,1,3696,8778,3850,462,10,11088,29106,15675,2772,144,1,36036,
%U A228337 99528,59202,12376,858,12,113256,335049,228085,60060,6240,209,1
%N A228337 Irregular triangle read by rows: the W-transformation of the Catalan triangle A033184.
%H A228337 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 A228337 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 A228337 Triangle begins:
%e A228337      1;
%e A228337      2;
%e A228337      4,    1;
%e A228337     10,    4;
%e A228337     20,   21,    1;
%e A228337     56,   70,    6;
%e A228337    140,  238,   50,    1;
%e A228337    420,  792,  210,    8;
%e A228337   1176, 2604,  990,   91,    1;
%e A228337   ...
%t A228337 nn = 12;
%t A228337 c[n_, k_] := If[k <= n, Binomial[2n-k, n] (k+1)/(n+1), 0];
%t A228337 a[n_, k_] := Table[c[If[OddQ[n], (n-1)/2+k+2i-2, n/2+k+i-1], 2k+j-1], {i, 1, 2}, {j, 1, 2}] // Permanent;
%t A228337 Table[a[n, k], {n, 0, nn}, {k, 0, n/2}] // Flatten (* _Jean-François Alcover_, Aug 12 2018 *)
%o A228337 (PARI) C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
%o A228337 matperm(M)=my(n=#M,t);sum(i=1,n!,t=numtoperm(n,i);prod(j=1,n,M[j,t[j]])); \\ from Rosetta code
%o A228337 W(n, k) = my(nn); if (n % 2, nn = (n-1)/2; matperm(matrix(2, 2, i, j, C(nn+k+2*i-2, 2*k+j-1))), nn = n/2; matperm(matrix(2, 2, i, j, C(nn+k+i-1, 2*k+j-1))));
%o A228337 aW(nn) = {for (n=0, nn, for (k=0, n\2, print1(W(n, k), ", ");); print(););} \\ _Michel Marcus_, Feb 13 2014
%Y A228337 Cf. A033184, A228334, A228335, A228336.
%K A228337 nonn,tabf
%O A228337 0,2
%A A228337 _N. J. A. Sloane_, Aug 26 2013
%E A228337 More terms from _Michel Marcus_, Feb 13 2014
%E A228337 A-number for Catalan triangle changed by _Michel Marcus_, Feb 13 2014