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).
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, 1, -9, -23, 19, 81, 41, 1, 1, 1, -11, -29, 49, 211, 141, 29, 1, 1, 1, -13, -35, 91, 401, 181, -363, -125, 1, 1, 1, -15, -41, 145, 651, 41, -2015, -1791, -365, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, ... 1, -1, -3, -5, -7, -9, -11, ... 1, -5, -11, -17, -23, -29, -35, ... 1, -5, 1, 19, 49, 91, 145, ... 1, 11, 81, 211, 401, 651, 961, ... 1, 41, 141, 181, 41, -399, -1259, ... 1, 29, -363, -2015, -5767, -12459, -22931, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
- T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Crossrefs
Programs
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Mathematica
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 *)
Formula
A(n,k) is the coefficient of x^n in the expansion of (1 + x - k*x^2)^n.
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).
n * A(n,k) = (2*n-1) * A(n-1,k) - (1+4*k) * (n-1) * A(n-2,k).