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A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.

%I A228336 #30 Nov 26 2018 06:59:24
%S A228336 1,1,1,2,2,1,4,6,3,1,10,15,12,4,1,25,45,36,20,5,1,70,126,126,70,30,6,
%T A228336 1,196,392,392,280,120,42,7,1,588,1176,1344,960,540,189,56,8,1,1764,
%U A228336 3780,4320,3600,2025,945,280,72,9,1,5544,11880,14850,12375,8250,3850,1540,396,90,10,1
%N A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.
%H A228336 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 A228336 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 A228336 Triangle begins:
%e A228336    1;
%e A228336    1,   1;
%e A228336    2,   2,   1;
%e A228336    4,   6,   3,  1;
%e A228336   10,  15,  12,  4,  1;
%e A228336   25,  45,  36, 20,  5, 1;
%e A228336   70, 126, 126, 70, 30, 6, 1;
%e A228336   ...
%t A228336 c[n_, k_] := Boole[k <= n] Binomial[2n - k, n] (k + 1)/(n + 1);
%t A228336 T[n_, k_] := Module[{nn, kk}, If[OddQ[n], nn = (n + 1)/2, nn = n/2]; If[OddQ[k], kk = (k - 1)/2, kk = k/2]; If [OddQ[n], If[OddQ[k], c[nn + kk, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk, 2kk + 1]], If[OddQ[k], c[nn + kk + 1, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk + 1, 2kk + 1]]]];
%t A228336 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 04 2018, from PARI *)
%o A228336 (PARI) C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
%o A228336 T(n, k) = {my(nn, kk); if (n % 2, nn = (n+1)/2, nn = n/2); if (k % 2, kk = (k-1)/2, kk = k/2); if ((n % 2), if (k % 2, C(nn+kk, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk, 2*kk+1)), if (k % 2, C(nn+kk+1, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk+1, 2*kk+1)));} \\ _Michel Marcus_, Feb 13 2014
%Y A228336 Cf. A033184, A228334, A228335, A228337.
%K A228336 nonn,tabl
%O A228336 0,4
%A A228336 _N. J. A. Sloane_, Aug 26 2013
%E A228336 More terms from _Michel Marcus_, Feb 13 2014
%E A228336 A-number for Catalan triangle changed by _Michel Marcus_, Feb 13 2014