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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).

%I A387466 #19 Aug 30 2025 10:12:56
%S A387466 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,
%T A387466 592,924,1,2,56,212,1096,2332,2624,3432,1,2,78,320,2566,7632,13964,
%U A387466 11776,12870,1,2,104,452,5320,20092,60864,83848,53344,48620
%N 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).
%F A387466 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.
%F A387466 A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
%F A387466 n*A(n,k) = 2*(2*n-1)*A(n-1,k) + 4*k^2*(n-1)*A(n-2,k) for n > 1.
%F A387466 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).
%F A387466 A(n,k) = [x^n] (1 + 2*x + (k^2+1)*x^2)^n.
%F A387466 E.g.f. of column k: exp(2*x) * BesselI(0, 2*sqrt(k^2+1)*x).
%e A387466 Square array begins:
%e A387466     1,    1,     1,     1,      1,      1,       1, ...
%e A387466     2,    2,     2,     2,      2,      2,       2, ...
%e A387466     6,    8,    14,    24,     38,     56,      78, ...
%e A387466    20,   32,    68,   128,    212,    320,     452, ...
%e A387466    70,  136,   406,  1096,   2566,   5320,   10006, ...
%e A387466   252,  592,  2332,  7632,  20092,  44752,   88092, ...
%e A387466   924, 2624, 13964, 60864, 210524, 607424, 1523724, ...
%o A387466 (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));
%Y A387466 Columns k=0..4 give A000984, A006139, A084770, A098455, A098456.
%Y A387466 Main diagonal gives A387467.
%Y A387466 Cf. A386621.
%K A387466 nonn,tabl,new
%O A387466 0,3
%A A387466 _Seiichi Manyama_, Aug 29 2025