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A144484 Triangle read by rows: T(n, k) = binomial(3*n+1-k, n-k) for n, k >= 0.

%I A144484 #8 May 13 2018 09:50:07
%S A144484 1,4,1,21,6,1,120,36,8,1,715,220,55,10,1,4368,1365,364,78,12,1,27132,
%T A144484 8568,2380,560,105,14,1,170544,54264,15504,3876,816,136,16,1,1081575,
%U A144484 346104,100947,26334,5985,1140,171,18,1,6906900,2220075,657800,177100
%N A144484 Triangle read by rows: T(n, k) = binomial(3*n+1-k, n-k) for n, k >= 0.
%C A144484 Previous name: A triangle sequence from a polynomial: p(x,n)=Sum[Binomial[3*n + 1 - m, n - m]*x^m, {m, 0, n}]; p(x,n)=Gamma[2*n+3]*Hypergeometric2F1[1,-n-1-3*n,x]/(Gamma[1+n]*Gamma[2+2*n}).
%D A144484 M. Jones, Further remarks on the enumeration of graphs, preprint, 2001.
%H A144484 Omer Egecioglu, Timothy Redmond, Charles Ryavec, <a href="http://arxiv.org/abs/0704.3398">Almost Product Evaluation of Hankel Determinants</a>, arXiv:0704.3398 [math.CO], 2007.
%F A144484 p(x,n)=Sum[Binomial[3*n + 1 - m, n - m]*x^m, {m, 0, n}];
%F A144484 p(x,n)=Gamma[2*n+3]*Hypergeometric2F1[1,-n-1-3*n,x]/(Gamma[1+n]*Gamma[2+2*n});
%e A144484 {1},
%e A144484 {4, 1},
%e A144484 {21, 6, 1},
%e A144484 {120, 36, 8, 1},
%e A144484 {715, 220, 55, 10, 1},
%e A144484 {4368, 1365, 364, 78, 12, 1},
%e A144484 {27132, 8568, 2380, 560, 105, 14, 1},
%e A144484 {170544, 54264, 15504, 3876, 816, 136, 16, 1},
%e A144484 {1081575, 346104, 100947, 26334, 5985, 1140, 171, 18, 1},
%e A144484 {6906900, 2220075, 657800, 177100, 42504, 8855, 1540, 210, 20, 1},
%e A144484 {44352165, 14307150, 4292145, 1184040, 296010, 65780, 12650, 2024, 253, 22, 1}
%t A144484 p[x_, n_] = Sum[Binomial[3*n + 1 - m, n - m]*x^m, {m, 0, n}]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]
%o A144484 (PARI) T(n, k) = binomial(3*n+1-k, n-k);
%o A144484 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, May 13 2018
%Y A144484 Cf. A025174 (row sums).
%K A144484 nonn,tabl
%O A144484 0,2
%A A144484 _Roger L. Bagula_, Oct 12 2008
%E A144484 New name from _Michel Marcus_, May 13 2018