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

1, 2, 4, 1, 10, 4, 20, 21, 1, 56, 70, 6, 140, 238, 50, 1, 420, 792, 210, 8, 1176, 2604, 990, 91, 1, 3696, 8778, 3850, 462, 10, 11088, 29106, 15675, 2772, 144, 1, 36036, 99528, 59202, 12376, 858, 12, 113256, 335049, 228085, 60060, 6240, 209, 1

Offset: 0

Author

N. J. A. Sloane, Aug 26 2013

Keywords

• nonn
• tabf

Examples

			Triangle begins:
     1;
     2;
     4,    1;
    10,    4;
    20,   21,    1;
    56,   70,    6;
   140,  238,   50,    1;
   420,  792,  210,    8;
  1176, 2604,  990,   91,    1;
  ...
		

Links

• Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
• Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Crossrefs

Cf. A033184, A228334, A228335, A228336.

Programs

Mathematica

nn = 12;
c[n_, k_] := If[k <= n, Binomial[2n-k, n] (k+1)/(n+1), 0];
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;
Table[a[n, k], {n, 0, nn}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

PARI

C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
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
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))));
aW(nn) = {for (n=0, nn, for (k=0, n\2, print1(W(n, k), ", ");); print(););} \\ Michel Marcus, Feb 13 2014

Extensions

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014