A279004 Irregular triangle read by rows: generalized Catalan triangle C_3(n,k).

1, 1, 1, 1, 2, 3, 3, 1, 3, 6, 9, 9, 1, 4, 10, 19, 28, 28, 1, 5, 15, 34, 62, 90, 90, 1, 6, 21, 55, 117, 207, 297, 297, 1, 7, 28, 83, 200, 407, 704, 1001, 1001, 1, 8, 36, 119, 319, 726, 1430, 2431, 3432, 3432

Offset: 0

Author

N. J. A. Sloane, Dec 07 2016

Keywords

• nonn
• tabf

Comments

The main diagonal is A000245, the third convolution of the Catalan numbers. See Tedford 2011. Also see A002057 for a similarly constructed triangle related to the fourth convolution of the Catalan numbers. - Peter Bala, Apr 14 2017

Examples

			Triangle begins:
1,1,1,
1,2,3,3,
1,3,6,9,9,
1,4,10,19,28,28,
1,5,15,34,62,90,90,
1,6,21,55,117,207,297,297,
1,7,28,83,200,407,704,1001,1001,
1,8,36,119,319,726,1430,2431,3432,3432,
...
		

Links

• Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
• S. J. Tedford, Combinatorial interpretations of convolutions of the Catalan numbers, Integers 11 (2011) #A3

Crossrefs

Cf. A009766, A000108, A000245, A002057.

Programs

Mathematica

c[m_][0, k_] /; k <= m-1 = 1;
c[m_][n_, k_] /; 0 <= k <= m+n-1 := c[m][n, k] = c[m][n-1, k]+c[m][n, k-1];
c[][, _] = 0;
Table[c[3][n, k], {n, 0, 7}, {k, 0, n+2}] // Flatten (* Jean-François Alcover, Oct 07 2018 *)