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a(n) = (2*n)!/a(n-1), with a(0)=1.
2

%I #17 Jul 13 2024 04:44:09

%S 1,2,12,60,672,5400,88704,982800,21288960,300736800,8089804800,

%T 138940401600,4465572249600,90311261040000,3375972620697600,

%U 78570797104800000,3348964839732019200,88156434351585600000,4219695698062344192000,123947946698329353600000

%N a(n) = (2*n)!/a(n-1), with a(0)=1.

%F a(n) ~ Gamma(1/4) * 2^(n - 1/4) * n^(n + 3/4) / exp(n) if n is even and a(n) ~ sqrt(Pi) * 2^(n + 9/4) * n^(n + 3/4) / (Gamma(1/4) * exp(n)) if n is odd. - _Vaclav Kotesovec_, Jul 09 2024

%F a(n) = Product_{k=0..floor((n-1)/2)} 2*(n-2*k)*(2*n-4*k-1). - _Andrew Howroyd_, Jul 09 2024

%t a[0] = 1; a[n_] := a[n] = (2 n)!/a[n - 1];

%t Table[a[n], {n, 0, 30}]

%t (* or *)

%t FullSimplify[Table[4^n * Gamma[3/4 + n/2] * Gamma[1 + n/2] / If[EvenQ[n], Sqrt[2]*Pi/Gamma[1/4], Sqrt[Pi]*Gamma[1/4]/4], {n, 0, 20}]] (* _Vaclav Kotesovec_, Jul 09 2024 *)

%o (PARI) a(n)={my(t=1); for(n=1, n, t = (2*n)!/t); t} \\ _Andrew Howroyd_, Jul 09 2024

%o (PARI) a(n)={prod(k=0, (n-1)\2, 2*(n-2*k)*(2*n-4*k-1))} \\ _Andrew Howroyd_, Jul 09 2024

%Y Cf. A372987, A372988.

%K nonn

%O 0,2

%A _Clark Kimberling_, Jul 09 2024