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A378240 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*n+3*r+k,n)/(3*n+3*r+k) for k > 0.

%I A378240 #35 Nov 21 2024 12:33:39
%S A378240 1,1,0,1,2,0,1,4,18,0,1,6,40,234,0,1,8,66,540,3570,0,1,10,96,926,8400,
%T A378240 59586,0,1,12,130,1400,14706,141876,1053570,0,1,14,168,1970,22720,
%U A378240 251622,2528760,19392490,0,1,16,210,2644,32690,394152,4524786,46815116,367677090,0
%N A378240 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*n+3*r+k,n)/(3*n+3*r+k) for k > 0.
%F A378240 G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
%F A378240 G.f. of column k: B(x)^k where B(x) is the g.f. of A364167.
%F A378240 B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+5). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+5) for n > 0.
%e A378240 Square array begins:
%e A378240   1,       1,       1,       1,       1,        1,        1, ...
%e A378240   0,       2,       4,       6,       8,       10,       12, ...
%e A378240   0,      18,      40,      66,      96,      130,      168, ...
%e A378240   0,     234,     540,     926,    1400,     1970,     2644, ...
%e A378240   0,    3570,    8400,   14706,   22720,    32690,    44880, ...
%e A378240   0,   59586,  141876,  251622,  394152,   575402,   801948, ...
%e A378240   0, 1053570, 2528760, 4524786, 7156128, 10553970, 14867704, ...
%o A378240 (PARI) T(n, k, t=3, 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 A378240 matrix(7, 7, n, k, T(n-1, k-1))
%Y A378240 Columns k=0..1 give A000007, A364167.
%Y A378240 Cf. A033877, A071949, A378236, A378237, A378238, A378239.
%K A378240 nonn,tabl
%O A378240 0,5
%A A378240 _Seiichi Manyama_, Nov 20 2024