A307819 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*k*x + k*(k+4)*x^2).
1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, 0, 5, 0, 1, -4, 3, 16, -5, 0, 1, -5, 8, 27, -56, -11, 0, 1, -6, 15, 32, -189, 48, 41, 0, 1, -7, 24, 25, -416, 567, 384, -29, 0, 1, -8, 35, 0, -725, 2176, 189, -1920, -125, 0, 1, -9, 48, -49, -1080, 5625, -4864, -11259, 3168, 365, 0
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 0, -1, -2, -3, -4, -5, -6, ... 0, -1, 0, 3, 8, 15, 24, ... 0, 5, 16, 27, 32, 25, 0, ... 0, -11, 48, 567, 2176, 5625, 11664, ... 0, 41, 384, 189, -4864, -24375, -74304, ... 0, -29, -1920, -11259, -23552, 9375, 228096, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
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Mathematica
A[n_, k_] := (-k)^n*Hypergeometric2F1[(1-n)/2, -n/2, 1, -4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)
Formula
A(n,k) is the coefficient of x^n in the expansion of (1 - k*x - k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = -k * (2*n-1) * A(n-1,k) - k * (k+4) * (n-1) * A(n-2,k).
Comments