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A378290 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+2*r+k,r) * binomial(r,n-r)/(n+2*r+k) for k > 0.

%I A378290 #13 Nov 22 2024 08:31:00
%S A378290 1,1,0,1,1,0,1,2,4,0,1,3,9,19,0,1,4,15,46,104,0,1,5,22,82,262,614,0,1,
%T A378290 6,30,128,486,1588,3816,0,1,7,39,185,789,3027,10053,24595,0,1,8,49,
%U A378290 254,1185,5052,19543,65686,162896,0,1,9,60,336,1689,7801,33290,129606,439658,1101922,0
%N A378290 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+2*r+k,r) * binomial(r,n-r)/(n+2*r+k) for k > 0.
%F A378290 G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + x * A_k(x)^(1/k)) )^k for k > 0.
%F A378290 G.f. of column k: B(x)^k where B(x) is the g.f. of A186997.
%F A378290 B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-2,k+3) for n > 1.
%e A378290 Square array begins:
%e A378290   1,    1,     1,     1,     1,     1,     1, ...
%e A378290   0,    1,     2,     3,     4,     5,     6, ...
%e A378290   0,    4,     9,    15,    22,    30,    39, ...
%e A378290   0,   19,    46,    82,   128,   185,   254, ...
%e A378290   0,  104,   262,   486,   789,  1185,  1689, ...
%e A378290   0,  614,  1588,  3027,  5052,  7801, 11430, ...
%e A378290   0, 3816, 10053, 19543, 33290, 52490, 78552, ...
%o A378290 (PARI) T(n, k, t=3, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
%o A378290 matrix(7, 7, n, k, T(n-1, k-1))
%Y A378290 Columns k=0..2 give A000007, A186997, A218045(n+2).
%Y A378290 Cf. A009766, A026300, A378289, A378291, A378292.
%K A378290 nonn,tabl
%O A378290 0,8
%A A378290 _Seiichi Manyama_, Nov 21 2024