cp's OEIS Frontend

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).

%I A386621 #44 Aug 29 2025 20:55:18
%S A386621 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,
%T A386621 592,20,1,12,152,1328,6216,10424,2624,0,1,14,218,2560,18886,68976,
%U A386621 80996,11776,70,1,16,296,4392,45256,276208,779456,637424,53344,0
%N 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).
%H A386621 Seiichi Manyama, <a href="/A386621/b386621.txt">Antidiagonals n = 0..139, flattened</a>
%F A386621 A(n,k) = Sum_{j=0..n} (k-i)^j * (k+i)^(n-j) * binomial(n,j)^2, where i is the imaginary unit.
%F A386621 A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
%F A386621 n*A(n,k) = 2*k*(2*n-1)*A(n-1,k) + 4*(n-1)*A(n-2,k) for n > 1.
%F A386621 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).
%F A386621 A(n,k) = [x^n] (1 + 2*k*x + (k^2+1)*x^2)^n.
%F A386621 E.g.f. of column k: exp(2*k*x) * BesselI(0, 2*sqrt(k^2+1)*x).
%e A386621 Square array begins:
%e A386621    1,    1,     1,      1,       1,        1,        1, ...
%e A386621    0,    2,     4,      6,       8,       10,       12, ...
%e A386621    2,    8,    26,     56,      98,      152,      218, ...
%e A386621    0,   32,   184,    576,    1328,     2560,     4392, ...
%e A386621    6,  136,  1366,   6216,   18886,    45256,    92886, ...
%e A386621    0,  592, 10424,  68976,  276208,   822800,  2020392, ...
%e A386621   20, 2624, 80996, 779456, 4114004, 15235520, 44758244, ...
%o A386621 (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));
%Y A386621 Columns k=0..3 give A126869, A006139, A098443, A387428.
%Y A386621 Main diagonal gives A387430.
%K A386621 nonn,tabl,new
%O A386621 0,5
%A A386621 _Seiichi Manyama_, Aug 29 2025