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Diagonal of rational function 1/(1 - (1 + x*y) * (x^2 + y^2)).
3

%I #23 Mar 23 2023 16:45:34

%S 1,0,2,4,8,24,56,144,376,960,2512,6560,17184,45248,119296,315392,

%T 835552,2217216,5893568,15687552,41810944,111567104,298016512,

%U 796832256,2132456704,5711486976,15309014528,41062927360,110213725184,295995574272,795391639552

%N Diagonal of rational function 1/(1 - (1 + x*y) * (x^2 + y^2)).

%H Seiichi Manyama, <a href="/A361726/b361726.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/sqrt(1 - 4 * x^2 * (1+x)^2).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*k,k) * binomial(2*k,n-2*k).

%F a(n) ~ (1 + sqrt(3))^(n + 1/2) / (2*3^(1/4)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 22 2023

%F n*a(n) = 2*(2*n-2)*a(n-2) + 4*(2*n-3)*a(n-3) + 2*(2*n-4)*a(n-4) for n > 3. - _Seiichi Manyama_, Mar 23 2023

%o (PARI) a(n) = sum(k=0, n\2, binomial(2*k, k)*binomial(2*k, n-2*k));

%Y Cf. A137635, A361727.

%Y Cf. A115962, A360266.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 22 2023