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a(n) = hypergeom([(1 - n)/2, -n/2], [2], 4*n^2).
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%I #7 Jan 08 2024 07:58:41

%S 1,1,5,28,609,6501,272701,4286815,272156417,5648748355,484054204501,

%T 12482361156398,1351553781736225,41650209565275195,

%U 5460281206077347469,195722005810272604876,30156361094764202326017,1232550298298392183231275,218366864894707599746619685

%N a(n) = hypergeom([(1 - n)/2, -n/2], [2], 4*n^2).

%F a(n) = p(n, n), where p(n, x) = hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2) are the Motzkin polynomials A359364.

%F a(n) ~ (exp(1) + (-1)^n) * 2^(n + 1/2) * n^(n - 3/2) / (sqrt(Pi) * exp(1/2)). - _Vaclav Kotesovec_, Jan 08 2024

%p a := n -> hypergeom([(1 - n)/2, -n/2], [2], 4*n^2):

%p seq(simplify(a(n)), n = 0..18);

%Y Cf. A359364.

%K nonn

%O 0,3

%A _Peter Luschny_, Jan 10 2023