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%I #12 Mar 27 2023 10:14:38
%S 1,1,3,1,3,18,1,3,15,126,1,3,12,90,945,1,3,9,57,585,7371,1,3,6,27,297,
%T 3969,58968,1,3,3,0,78,1629,27657,480168,1,3,0,-24,-75,207,9216,
%U 196290,3961386,1,3,-3,-45,-165,-438,459,53217,1411965,33011550
%N Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 - x)^k)^(1/3).
%F n*T(n,k) = 3 * Sum_{j=0..k} (-1)^j * binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.
%F T(n,k) = (-1)^n * Sum_{j=0..n} 9^j * binomial(-1/3,j) * binomial(k*j,n-j).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 3, 3, 3, 3, 3, 3, ...
%e 18, 15, 12, 9, 6, 3, ...
%e 126, 90, 57, 27, 0, -24, ...
%e 945, 585, 297, 78, -75, -165, ...
%e 7371, 3969, 1629, 207, -438, -444, ...
%o (PARI) T(n, k) = (-1)^n*sum(j=0, n, 9^j*binomial(-1/3, j)*binomial(k*j, n-j));
%Y Columns k=0..3 give A004987, A361843, A361844, A361845.
%Y Main diagonal gives A361847.
%Y Cf. A361834, A361839.
%K sign,tabl
%O 0,3
%A _Seiichi Manyama_, Mar 26 2023
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