A362125 - OEIS

%I #12 Apr 09 2023 08:08:01

%S 1,1,0,1,1,0,1,2,2,0,1,3,7,3,0,1,4,15,18,5,0,1,5,26,55,47,8,0,1,6,40,

%T 124,198,118,13,0,1,7,57,235,571,681,290,21,0,1,8,77,398,1320,2500,

%U 2263,702,34,0,1,9,100,623,2640,7026,10504,7341,1677,55,0

%N Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - x*(1+x)^k)^k.

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

%e Square array begins:

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

%e   0, 1,   2,   3,    4,    5, ...

%e   0, 2,   7,  15,   26,   40, ...

%e   0, 3,  18,  55,  124,  235, ...

%e   0, 5,  47, 198,  571, 1320, ...

%e   0, 8, 118, 681, 2500, 7026, ...

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

%Y Columns k=0..2 give A000007, A000045(n+1), A362126.

%Y Main diagonal gives A362080.

%Y Cf. A362078, A362079.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Apr 08 2023