A333989 - OEIS

%I #17 Sep 04 2020 15:20:26

%S 1,1,1,1,0,1,1,-1,-4,1,1,-2,-7,0,1,1,-3,-8,23,16,1,1,-4,-7,64,17,0,1,

%T 1,-5,-4,117,-128,-241,-64,1,1,-6,1,176,-527,-512,329,0,1,1,-7,8,235,

%U -1264,237,4096,1511,256,1,1,-8,17,288,-2399,3776,11753,-8192,-5983,0,1

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1+(k-1)*x) / (1+2*(k-1)*x+((k+1)*x)^2).

%F T(n,k) = Sum_{j=0..n} (-k)^j * binomial(2*n,2*j).

%F T(0,k) = 1, T(1,k) = 1-k and T(n,k) = -2 * (k-1) * T(n-1,k) - (k+1)^2 * T(n-2,k) for n>1.

%e Square array begins:

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

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

%e   1, -4,   -7,   -8,   -7,    -4, ...

%e   1,  0,   23,   64,  117,   176, ...

%e   1, 16,   17, -128, -527, -1264, ...

%e   1,  0, -241, -512,  237,  3776, ...

%t T[n_, 0] := 1; T[n_, k_] := Sum[(-k)^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Sep 04 2020 *)

%o (PARI) {T(n, k) = sum(j=0, n, (-k)^j*binomial(2*n, 2*j))}

%Y Main diagonal gives A333991.

%Y Cf. A307884, A333988.

%K sign,tabl

%O 0,9

%A _Seiichi Manyama_, Sep 04 2020