A308035 - OEIS

%I #25 May 12 2021 03:56:48

%S 1,1,1,1,1,1,1,1,0,1,1,1,-1,-2,1,1,1,-2,-5,-3,1,1,1,-3,-8,-3,1,1,1,1,

%T -4,-11,1,21,11,1,1,1,-5,-14,9,61,51,15,1,1,1,-6,-17,21,121,91,-41,

%U -13,1,1,1,-7,-20,37,201,101,-377,-391,-77,1

%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1+4*k)*x^2)).

%H Seiichi Manyama, <a href="/A308035/b308035.txt">Antidiagonals n = 0..139, flattened</a>

%F A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 + x - k*x^2)^(n+1).

%F A(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,2*j) * A000108(j).

%F (n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1+4*k) * (n-1) * A(n-2,k).

%e Square array begins:

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

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

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

%e    1, -2,  -5,   -8,   -11,   -14,   -17, ...

%e    1, -3,  -3,    1,     9,    21,    37, ...

%e    1,  1,  21,   61,   121,   201,   301, ...

%e    1, 11,  51,   91,   101,    51,   -89, ...

%e    1, 15, -41, -377, -1203, -2729, -5165, ...

%t T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 12 2021 *)

%Y Columns k=2..3 give (-1)^n * A091593, A308036.

%Y Main diagonal gives A307947.

%Y Cf. A000108, A306684, A307860.

%K sign,tabl

%O 0,14

%A _Seiichi Manyama_, May 10 2019