A386621 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4*k*x - 4*x^2).
1, 1, 0, 1, 2, 2, 1, 4, 8, 0, 1, 6, 26, 32, 6, 1, 8, 56, 184, 136, 0, 1, 10, 98, 576, 1366, 592, 20, 1, 12, 152, 1328, 6216, 10424, 2624, 0, 1, 14, 218, 2560, 18886, 68976, 80996, 11776, 70, 1, 16, 296, 4392, 45256, 276208, 779456, 637424, 53344, 0
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 2, 4, 6, 8, 10, 12, ... 2, 8, 26, 56, 98, 152, 218, ... 0, 32, 184, 576, 1328, 2560, 4392, ... 6, 136, 1366, 6216, 18886, 45256, 92886, ... 0, 592, 10424, 68976, 276208, 822800, 2020392, ... 20, 2624, 80996, 779456, 4114004, 15235520, 44758244, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Programs
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PARI
a(n, k) = sum(j=0, n\2, (k^2+1)^j*(2*k)^(n-2*j)*binomial(n, 2*j)*binomial(2*j, j));
Formula
A(n,k) = Sum_{j=0..n} (k-i)^j * (k+i)^(n-j) * binomial(n,j)^2, where i is the imaginary unit.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
n*A(n,k) = 2*k*(2*n-1)*A(n-1,k) + 4*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} (k^2+1)^j * (2*k)^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
A(n,k) = [x^n] (1 + 2*k*x + (k^2+1)*x^2)^n.
E.g.f. of column k: exp(2*k*x) * BesselI(0, 2*sqrt(k^2+1)*x).