A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4*x - (2*k*x)^2).
1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 14, 32, 70, 1, 2, 24, 68, 136, 252, 1, 2, 38, 128, 406, 592, 924, 1, 2, 56, 212, 1096, 2332, 2624, 3432, 1, 2, 78, 320, 2566, 7632, 13964, 11776, 12870, 1, 2, 104, 452, 5320, 20092, 60864, 83848, 53344, 48620
Offset: 0
Examples
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, 2, ...
6, 8, 14, 24, 38, 56, 78, ...
20, 32, 68, 128, 212, 320, 452, ...
70, 136, 406, 1096, 2566, 5320, 10006, ...
252, 592, 2332, 7632, 20092, 44752, 88092, ...
924, 2624, 13964, 60864, 210524, 607424, 1523724, ...
Crossrefs
Programs
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PARI
a(n, k) = sum(j=0, n\2, (k^2+1)^j*2^(n-2*j)*binomial(n, 2*j)*binomial(2*j, j));
Formula
A(n,k) = Sum_{j=0..n} (1-k*i)^j * (1+k*i)^(n-j) * binomial(n,j)^2, where i is the imaginary unit.
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
n*A(n,k) = 2*(2*n-1)*A(n-1,k) + 4*k^2*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} (k^2+1)^j * 2^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
A(n,k) = [x^n] (1 + 2*x + (k^2+1)*x^2)^n.
E.g.f. of column k: exp(2*x) * BesselI(0, 2*sqrt(k^2+1)*x).