

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
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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.
KyuHwan Lee, Sejin 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 <= m1 = 1;
c[m_][n_, k_] /; 0 <= k <= m+n1 := c[m][n, k] = c[m][n1, k]+c[m][n, k1];
c[_][_, _] = 0;
Table[c[3][n, k], {n, 0, 7}, {k, 0, n+2}] // Flatten (* JeanFrançois Alcover, Oct 07 2018 *)


CROSSREFS

Cf. A009766, A000108, A000245, A002057.
Sequence in context: A256909 A343533 A308734 * A172528 A087074 A136453
Adjacent sequences: A279001 A279002 A279003 * A279005 A279006 A279007


KEYWORD

nonn,tabf


AUTHOR

N. J. A. Sloane, Dec 07 2016


STATUS

approved



