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%I #15 Nov 24 2024 09:26:46
%S 1,1,0,1,2,0,1,4,6,0,1,6,16,30,0,1,8,30,84,170,0,1,10,48,170,496,1050,
%T 0,1,12,70,296,1050,3140,6846,0,1,14,96,470,1920,6846,20832,46374,0,1,
%U 16,126,700,3210,12936,46374,142932,323154,0,1,18,160,994,5040,22402,89712,323154,1005856,2301618,0
%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*r+k,n)/(3*r+k) for k > 0.
%F G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x * A_k(x)^(3/k) )^k for k > 0.
%F G.f. of column k: B(x)^k where B(x) is the g.f. of A366266.
%F B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k+2) for n > 0.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 2, 4, 6, 8, 10, 12, ...
%e 0, 6, 16, 30, 48, 70, 96, ...
%e 0, 30, 84, 170, 296, 470, 700, ...
%e 0, 170, 496, 1050, 1920, 3210, 5040, ...
%e 0, 1050, 3140, 6846, 12936, 22402, 36492, ...
%e 0, 6846, 20832, 46374, 89712, 159390, 266800, ...
%o (PARI) T(n, k, t=0, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
%o matrix(7, 7, n, k, T(n-1, k-1))
%Y Columns k=0..1 give A000007, A366266.
%Y Main diagonal gives A378378.
%Y Cf. A266213, A378317.
%Y Cf. A378323.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, Nov 23 2024