OFFSET
0,5
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
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
EXAMPLE
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,
...
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Dec 07 2016
STATUS
approved