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A279004 Irregular triangle read by rows: generalized Catalan triangle C_3(n,k). 2
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..51.

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

Cf. A009766, A000108, A000245, A002057.

Sequence in context: A109199 A279813 A256909 * A172528 A087074 A136453

Adjacent sequences:  A279001 A279002 A279003 * A279005 A279006 A279007

KEYWORD

nonn,tabf,changed

AUTHOR

N. J. A. Sloane, Dec 07 2016

STATUS

approved

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Last modified October 15 15:28 EDT 2018. Contains 316236 sequences. (Running on oeis4.)