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