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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).
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%I #12 Mar 27 2023 10:14:42

%S 1,1,3,1,3,18,1,3,21,126,1,3,24,162,945,1,3,27,201,1341,7371,1,3,30,

%T 243,1809,11529,58968,1,3,33,288,2352,16893,101619,480168,1,3,36,336,

%U 2973,23607,161676,911466,3961386,1,3,39,387,3675,31818,242757,1574289,8281737,33011550

%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} binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.

%F T(n,k) = 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, 21, 24, 27, 30, 33, ...

%e 126, 162, 201, 243, 288, 336, ...

%e 945, 1341, 1809, 2352, 2973, 3675, ...

%e 7371, 11529, 16893, 23607, 31818, 41676, ...

%o (PARI) T(n,k) = sum(j=0, n, (-9)^j*binomial(-1/3, j)*binomial(k*j, n-j));

%Y Columns k=0..3 give A004987, A180400, A361841, A361842.

%Y Main diagonal gives A361846.

%Y Cf. A361830, A361840.

%K nonn,tabl

%O 0,3

%A _Seiichi Manyama_, Mar 26 2023