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A324568 a(n) = Sum_{i=0..n, j=0..n} (binomial(2*i, j) + binomial(2*j, i)).

%I A324568 #7 Mar 03 2025 13:35:20
%S A324568 2,8,32,124,482,1882,7380,29036,114530,452638,1791638,7100430,
%T A324568 28167986,111837902,444351292,1766536044,7026526226,27960911422,
%U A324568 111308958942,443258277254,1765690504666,7035402933402,28039342445582,111773962249054,445654589001882
%N A324568 a(n) = Sum_{i=0..n, j=0..n} (binomial(2*i, j) + binomial(2*j, i)).
%H A324568 Robert Israel, <a href="/A324568/b324568.txt">Table of n, a(n) for n = 0..1653</a>
%F A324568 Recurrence: 2*(n+1)*(5*n^2 - 21*n + 20)*a(n) = (85*n^3 - 332*n^2 + 243*n + 100)*a(n-1) - 3*(65*n^3 - 298*n^2 + 377*n - 100)*a(n-2) + 2*(20*n^3 - 109*n^2 + 191*n - 100)*a(n-3) + 8*(2*n - 5)*(5*n^2 - 11*n + 4)*a(n-4).
%F A324568 a(n) ~ 4^(n+1)/3 * (1 + 5/(3*sqrt(Pi*n))).
%p A324568 R:= 2: r:= 2;
%p A324568 for n from 1 to 30 do
%p A324568   v:= 2*binomial(2*n,n) + 2*add(binomial(2*n,j),j=0..n-1) + 2*add(binomial(2*j,n),j=ceil(n/2) .. n-1);
%p A324568   r:= r+v;
%p A324568   R:= R,r;
%p A324568 od:
%p A324568 R; # _Robert Israel_, Mar 02 2025
%t A324568 Table[Sum[Binomial[2*i, j] + Binomial[2*j, i], {i, 0, n}, {j, 0, n}], {n, 0, 30}]
%Y A324568 Cf. A000984, A006134, A007685, A324566, A324567.
%K A324568 nonn
%O A324568 0,1
%A A324568 _Vaclav Kotesovec_, Mar 07 2019