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%I #13 Jan 27 2018 06:46:03
%S 1,0,1,2,0,3,0,12,0,10,10,0,60,0,35,0,105,0,280,0,126,56,0,756,0,1260,
%T 0,462,0,840,0,4620,0,5544,0,1716,330,0,7920,0,25740,0,24024,0,6435,0,
%U 6435,0,60060,0,135135,0,102960,0,24310
%N Triangle read by rows, the unsigned coefficients of G(n, n, x/2) where G(n,a,x) denotes the n-th Gegenbauer polynomial, T(n, k) for 0 <= k <= n.
%F G(n, x) = binomial(3*n-1, n)*hypergeom([-n, 3*n], [n+1/2], 1/2 - x/4).
%e [0] 1
%e [1] 0, 1
%e [2] 2, 0, 3
%e [3] 0, 12, 0, 10
%e [4] 10, 0, 60, 0, 35
%e [5] 0, 105, 0, 280, 0, 126
%e [6] 56, 0, 756, 0, 1260, 0, 462
%e [7] 0, 840, 0, 4620, 0, 5544, 0, 1716
%e [8] 330, 0, 7920, 0, 25740, 0, 24024, 0, 6435
%e [9] 0, 6435, 0, 60060, 0, 135135, 0, 102960, 0, 24310
%p with(orthopoly):
%p seq(seq((-1)^iquo(n-k, 2)*coeff(G(n,n,x/2),x,k), k=0..n), n=0..9);
%t p[n_] := Binomial[3 n - 1, n] Hypergeometric2F1[-n, 3 n, n + 1/2, 1/2 - x/4];
%t Flatten[Table[(-1)^Floor[(n-k)/2] Coefficient[p[n], x, k], {n,0,9}, {k,0,n}]]
%Y T(2n, 0) = A165817(n). T(n,n) = A088218(n). Row sums are A213684.
%Y Cf. A109187.
%K nonn,tabl
%O 0,4
%A _Peter Luschny_, Jan 25 2018
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