A380912 Two-Catalan Triangle read by rows, for n>=0 and k>=0.

1, 1, 1, 1, 3, 6, 6, 3, 1, 15, 36, 40, 29, 15, 5, 1, 91, 232, 280, 238, 154, 76, 28, 7, 1, 603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1, 4213, 11298, 15026, 14938, 12078, 8162, 4642, 2211, 869, 274, 66, 11, 1, 30537, 83097, 113841, 118482, 102102, 75075, 47619, 26091, 12285, 4914, 1638, 441, 91, 13, 1

Offset: 0

Author

Michel Marcus, Feb 08 2025

Keywords

• nonn
• tabf

Comments

It appears that this triangle is made of the even rows of A089942.

Examples

			Triangle begins:
  1;
  1, 1, 1;
  3, 6, 6, 3, 1;
  15, 36, 40, 29, 15, 5, 1;
  91, 232, 280, 238, 154, 76, 28, 7, 1;
  603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1;
  ...
		

Links

• Yousra Ghemit and Moussa Ahmia, Two-Catalan numbers: combinatorial interpretation and log-convexity, UPB Sci. Bull., Series A (2024) Vol 86, Iss. 4. See Table 2 p. 94.

Crossrefs

Cf. A027907, A089942, A380899.

Programs

PARI

t(n, k) = polcoef((1 + x + x^2)^n, k); \\ A027907
T(n, k) = t(2*n, 2*n+k) - t(2*n, 2*n+k+1);
row(n) = vector(2*n+1, k, T(n,k-1));

Formula

T(n,k) = A027907(2*n,k) - A027907(2*n,k+1).