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

%I A279004 #13 Oct 07 2018 08:02:08
%S A279004 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 A279004 117,207,297,297,1,7,28,83,200,407,704,1001,1001,1,8,36,119,319,726,
%U A279004 1430,2431,3432,3432
%N A279004 Irregular triangle read by rows: generalized Catalan triangle C_3(n,k).
%C A279004 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 A279004 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 A279004 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 A279004 Triangle begins:
%e A279004 1,1,1,
%e A279004 1,2,3,3,
%e A279004 1,3,6,9,9,
%e A279004 1,4,10,19,28,28,
%e A279004 1,5,15,34,62,90,90,
%e A279004 1,6,21,55,117,207,297,297,
%e A279004 1,7,28,83,200,407,704,1001,1001,
%e A279004 1,8,36,119,319,726,1430,2431,3432,3432,
%e A279004 ...
%t A279004 c[m_][0, k_] /; k <= m-1 = 1;
%t A279004 c[m_][n_, k_] /; 0 <= k <= m+n-1 := c[m][n, k] = c[m][n-1, k]+c[m][n, k-1];
%t A279004 c[_][_, _] = 0;
%t A279004 Table[c[3][n, k], {n, 0, 7}, {k, 0, n+2}] // Flatten (* _Jean-François Alcover_, Oct 07 2018 *)
%Y A279004 Cf. A009766, A000108, A000245, A002057.
%K A279004 nonn,tabf
%O A279004 0,5
%A A279004 _N. J. A. Sloane_, Dec 07 2016