%I #11 Apr 08 2023 11:06:58
%S 1,1,0,1,1,0,1,2,3,0,1,3,7,10,0,1,4,12,28,45,0,1,5,18,55,145,251,0,1,
%T 6,25,92,315,896,1624,0,1,7,33,140,571,2106,6328,11908,0,1,8,42,200,
%U 930,4076,15946,50212,97545,0,1,9,52,273,1410,7026,32718,134730,441489,880660,0
%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^n)^k.
%F T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-k,j) * binomial(n*j,n-j) = Sum_{j=0..n} binomial(j+k-1,j) * binomial(n*j,n-j).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 3, 7, 12, 18, 25, ...
%e 0, 10, 28, 55, 92, 140, ...
%e 0, 45, 145, 315, 571, 930, ...
%e 0, 251, 896, 2106, 4076, 7026, ...
%o (PARI) T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(n*j, n-j));
%Y Columns k=0..3 give A000007, A099237, A362084, A362085.
%Y Main diagonal gives A362080.
%Y Cf. A362078.
%K nonn,tabl
%O 0,8
%A _Seiichi Manyama_, Apr 08 2023