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A298610 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.

%I A298610 #13 Jan 27 2018 06:46:03
%S A298610 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 A298610 0,462,0,840,0,4620,0,5544,0,1716,330,0,7920,0,25740,0,24024,0,6435,0,
%U A298610 6435,0,60060,0,135135,0,102960,0,24310
%N A298610 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 A298610 G(n, x) = binomial(3*n-1, n)*hypergeom([-n, 3*n], [n+1/2], 1/2 - x/4).
%e A298610 [0]   1
%e A298610 [1]   0,    1
%e A298610 [2]   2,    0,    3
%e A298610 [3]   0,   12,    0,    10
%e A298610 [4]  10,    0,   60,     0,    35
%e A298610 [5]   0,  105,    0,   280,     0,    126
%e A298610 [6]  56,    0,  756,     0,  1260,      0,   462
%e A298610 [7]   0,  840,    0,  4620,     0,   5544,     0,   1716
%e A298610 [8] 330,    0, 7920,     0, 25740,      0, 24024,      0, 6435
%e A298610 [9]   0, 6435,    0, 60060,     0, 135135,     0, 102960,    0,  24310
%p A298610 with(orthopoly):
%p A298610 seq(seq((-1)^iquo(n-k, 2)*coeff(G(n,n,x/2),x,k), k=0..n), n=0..9);
%t A298610 p[n_] := Binomial[3 n - 1, n] Hypergeometric2F1[-n, 3 n, n + 1/2, 1/2 - x/4];
%t A298610 Flatten[Table[(-1)^Floor[(n-k)/2] Coefficient[p[n], x, k], {n,0,9}, {k,0,n}]]
%Y A298610 T(2n, 0) = A165817(n). T(n,n) = A088218(n). Row sums are A213684.
%Y A298610 Cf. A109187.
%K A298610 nonn,tabl
%O A298610 0,4
%A A298610 _Peter Luschny_, Jan 25 2018