%I #48 Oct 31 2022 15:24:16

%S 1,1,0,1,2,0,1,4,3,0,1,6,16,4,0,1,8,39,64,5,0,1,10,72,258,256,6,0,1,

%T 12,115,664,1719,1024,7,0,1,14,168,1360,6184,11496,4096,8,0,1,16,231,

%U 2424,16265,57888,77052,16384,9,0,1,18,304,3934,35400,195660,543544,517194,65536,10,0

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j) * binomial(k*(n-j),n-j).

%F T(n,k) = Sum_{j=0..n} (k-1)^(n-j) * binomial(k*n+1,j).

%F T(n,k) = Sum_{j=0..n} k^(n-j) * binomial((k-1)*n+j,j).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, 2, 4, 6, 8, 10, ...

%e 0, 3, 16, 39, 72, 115, ...

%e 0, 4, 64, 258, 664, 1360, ...

%e 0, 5, 256, 1719, 6184, 16265, ...

%e 0, 6, 1024, 11496, 57888, 195660, ...

%o (PARI) T(n, k) = sum(j=0, n, binomial(k*j, j)*binomial(k*(n-j), n-j));

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

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

%Y Column k=0-7 give: A000007, A001477(n+1), A000302, A006256, A078995, A079678, A079679, A079563.

%Y Main diagonal gives A358145.

%Y Cf. A358146.

%K nonn,tabl

%O 0,5

%A _Seiichi Manyama_, Oct 31 2022