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A362079 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.

%I A362079 #11 Apr 08 2023 11:06:58
%S A362079 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 A362079 6,25,92,315,896,1624,0,1,7,33,140,571,2106,6328,11908,0,1,8,42,200,
%U A362079 930,4076,15946,50212,97545,0,1,9,52,273,1410,7026,32718,134730,441489,880660,0
%N A362079 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 A362079 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 A362079 Square array begins:
%e A362079   1,   1,   1,    1,    1,    1, ...
%e A362079   0,   1,   2,    3,    4,    5, ...
%e A362079   0,   3,   7,   12,   18,   25, ...
%e A362079   0,  10,  28,   55,   92,  140, ...
%e A362079   0,  45, 145,  315,  571,  930, ...
%e A362079   0, 251, 896, 2106, 4076, 7026, ...
%o A362079 (PARI) T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(n*j, n-j));
%Y A362079 Columns k=0..3 give A000007, A099237, A362084, A362085.
%Y A362079 Main diagonal gives A362080.
%Y A362079 Cf. A362078.
%K A362079 nonn,tabl
%O A362079 0,8
%A A362079 _Seiichi Manyama_, Apr 08 2023