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A361752 a(n) = Sum_{k=0..floor(n/2)} binomial(2*(n-2*k),k) * binomial(2*(n-2*k),n-2*k).

%I A361752 #20 Mar 23 2023 19:31:03
%S A361752 1,2,6,24,94,374,1520,6252,25942,108408,455586,1923444,8151856,
%T A361752 34661252,147788484,631660788,2705471254,11609393084,49899207640,
%U A361752 214792704256,925811868178,3995288307392,17260287754284,74641620619072,323080683587056,1399606566298916
%N A361752 a(n) = Sum_{k=0..floor(n/2)} binomial(2*(n-2*k),k) * binomial(2*(n-2*k),n-2*k).
%C A361752 Diagonal of rational function 1/(1 - (1 + (x*y)^2) * (x + y)).
%H A361752 Seiichi Manyama, <a href="/A361752/b361752.txt">Table of n, a(n) for n = 0..1000</a>
%F A361752 G.f.: 1/sqrt(1 - 4*x*(1 + x^2)^2).
%F A361752 n*a(n) = 2*(2*n-1)*a(n-1) + 4*(2*n-3)*a(n-3) + 2*(2*n-5)*a(n-5) for n > 4.
%o A361752 (PARI) a(n) = sum(k=0, n\2, binomial(2*(n-2*k), k)*binomial(2*(n-2*k), n-2*k));
%o A361752 (Python)
%o A361752 from math import comb
%o A361752 def A361752(n): return sum(comb(m:=(r:=n-(k<<1))<<1,k)*comb(m,r) for k in range((n>>1)+1)) # _Chai Wah Wu_, Mar 23 2023
%Y A361752 Cf. A137635, A361753.
%Y A361752 Cf. A360266.
%K A361752 nonn
%O A361752 0,2
%A A361752 _Seiichi Manyama_, Mar 23 2023