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%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
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