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a(n) = Sum_{k=0..floor(n^2/(2*n+1))} binomial(n * (n-2*k),k).
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%I #8 Jun 15 2024 09:23:06

%S 1,1,1,4,9,26,91,281,1105,4105,16576,70643,301405,1382928,6363876,

%T 30605836,150820769,758835104,3941917840,20787546715,112615930451,

%U 620969188400,3492709446326,20034747631656,116780977502105,693539635192626,4181549476945504,25627647913369903

%N a(n) = Sum_{k=0..floor(n^2/(2*n+1))} binomial(n * (n-2*k),k).

%F a(n) = [x^n] 1/(1 - x * (1 + x^2)^n).

%o (PARI) a(n) = sum(k=0, n^2\(2*n+1), binomial(n*(n-2*k), k));

%Y Main diagonal of A373717.

%Y Cf. A099237.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jun 15 2024