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Irregular triangle read by rows: generalized Catalan triangle C_3(n,k).
2

%I #13 Oct 07 2018 08:02:08

%S 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,

%T 117,207,297,297,1,7,28,83,200,407,704,1001,1001,1,8,36,119,319,726,

%U 1430,2431,3432,3432

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

%C 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

%H Kyu-Hwan Lee, Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.

%H S. J. Tedford, <a href="http://www.emis.de/journals/INTEGERS/papers/l3/l3.Abstract.html">Combinatorial interpretations of convolutions of the Catalan numbers</a>, Integers 11 (2011) #A3

%e Triangle begins:

%e 1,1,1,

%e 1,2,3,3,

%e 1,3,6,9,9,

%e 1,4,10,19,28,28,

%e 1,5,15,34,62,90,90,

%e 1,6,21,55,117,207,297,297,

%e 1,7,28,83,200,407,704,1001,1001,

%e 1,8,36,119,319,726,1430,2431,3432,3432,

%e ...

%t c[m_][0, k_] /; k <= m-1 = 1;

%t c[m_][n_, k_] /; 0 <= k <= m+n-1 := c[m][n, k] = c[m][n-1, k]+c[m][n, k-1];

%t c[_][_, _] = 0;

%t Table[c[3][n, k], {n, 0, 7}, {k, 0, n+2}] // Flatten (* _Jean-François Alcover_, Oct 07 2018 *)

%Y Cf. A009766, A000108, A000245, A002057.

%K nonn,tabf

%O 0,5

%A _N. J. A. Sloane_, Dec 07 2016