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A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1+4*k)*x^2).

%I A307860 #27 May 13 2021 02:36:00
%S A307860 1,1,1,1,1,1,1,1,-1,1,1,1,-3,-5,1,1,1,-5,-11,-5,1,1,1,-7,-17,1,11,1,1,
%T A307860 1,-9,-23,19,81,41,1,1,1,-11,-29,49,211,141,29,1,1,1,-13,-35,91,401,
%U A307860 181,-363,-125,1,1,1,-15,-41,145,651,41,-2015,-1791,-365,1
%N A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1+4*k)*x^2).
%H A307860 Seiichi Manyama, <a href="/A307860/b307860.txt">Antidiagonals n = 0..139, flattened</a>
%H A307860 T. D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F A307860 A(n,k) is the coefficient of x^n in the expansion of (1 + x - k*x^2)^n.
%F A307860 A(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,2*j) * binomial(2*j,j).
%F A307860 n * A(n,k) = (2*n-1) * A(n-1,k) - (1+4*k) * (n-1) * A(n-2,k).
%e A307860 Square array begins:
%e A307860    1,  1,    1,     1,     1,      1,      1, ...
%e A307860    1,  1,    1,     1,     1,      1,      1, ...
%e A307860    1, -1,   -3,    -5,    -7,     -9,    -11, ...
%e A307860    1, -5,  -11,   -17,   -23,    -29,    -35, ...
%e A307860    1, -5,    1,    19,    49,     91,    145, ...
%e A307860    1, 11,   81,   211,   401,    651,    961, ...
%e A307860    1, 41,  141,   181,    41,   -399,  -1259, ...
%e A307860    1, 29, -363, -2015, -5767, -12459, -22931, ...
%t A307860 T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 13 2021 *)
%Y A307860 Columns k=0..5 give A000012, A098331, A098332, A098333, A098334.
%Y A307860 Main diagonal gives A307862.
%Y A307860 Cf. A307819, A307855.
%K A307860 sign,tabl
%O A307860 0,13
%A A307860 _Seiichi Manyama_, May 02 2019