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

0,2

Links

Table of n, a(n) for n=0..48.

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.

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

Cf. A033184, A228334, A228335, A228336.

Sequence in context: A363575 A271206 A211244 * A114506 A114848 A135330

Adjacent sequences: A228334 A228335 A228336 * A228338 A228339 A228340

Keyword

nonn,tabf

Author

N. J. A. Sloane, Aug 26 2013

Extensions

More terms from Michel Marcus, Feb 13 2014

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

Status

approved