A380899 Three-Catalan Triangle read by rows, for n>=0 and k>=0.

1, 1, 1, 1, 1, 4, 9, 11, 10, 6, 3, 1, 34, 90, 120, 120, 96, 64, 35, 15, 5, 1, 364, 1000, 1400, 1505, 1351, 1044, 700, 406, 202, 84, 28, 7, 1, 4269, 11925, 17225, 19425, 18657, 15753, 11845, 7965, 4785, 2553, 1197, 485, 165, 45, 9, 1

Offset: 0

Author

Michel Marcus, Feb 07 2025

Keywords

• nonn
• tabf

Examples

			Triangle begins:
   1
   1    1    1    1
   4    9   11   10    6    3   1
  34   90  120  120   96   64  35  15   5  1
 364 1000 1400 1505 1351 1044 700 406 202 84 28 7 1
 ...
		

Links

• Boualam Rezig and Moussa Ahmia, Combinatorics of three-Catalan numbers and some positivities, arXiv:2502.03615 [math.CO], 2025. See Table 2 p. 5. T(4,7)=405 is a typo.

Crossrefs

Row sums are A005721.

Cf. A008287.

Programs

PARI

t(n, k) = polcoef((1 + x + x^2 + x^3)^n, k); \\ A008287
T(n, k) = t(2*n, 3*n+k) - t(2*n, 3*n+k+1);
row(n) = vector(3*n+1, k, T(n,k-1));

Formula

T(n, k) = A008287(2*n, 3*n+k) - A008287(2*n, 3*n+k+1).