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A330793 a(n) = A193737(2*n, n).

%I A330793 #14 Oct 24 2023 04:28:31
%S A330793 1,2,8,36,170,826,4088,20496,103752,529100,2714140,13989560,72393412,
%T A330793 375877684,1957199120,10216355632,53443289946,280101010170,
%U A330793 1470508417340,7731675774900,40706787482130,214580612067690,1132389348358320,5981916549623040,31629125981208600
%N A330793 a(n) = A193737(2*n, n).
%H A330793 G. C. Greubel, <a href="/A330793/b330793.txt">Table of n, a(n) for n = 0..1000</a>
%F A330793 D-finite with recurrence a(n) = ( 2*(11*n-3)*(n-1)*a(n-1) + 3*(3*n - 4)*(3*n-5)*a(n-2) )/(5*n*(n-1)).
%F A330793 a(n) = [x^n] (16 + 8*hypergeometric2F1([2/3, 1/3], [1/2], (1+x)*27/32) + sqrt(18*(1+x))* hypergeometric2F1([7/6, 5/6], [3/2], (1+x)*27/32))/48.
%F A330793 a(n) = [x^n] (1/(3*sqrt(5 - 27*x)))*(sqrt(5 - 27*x) + 2*sqrt(2)*cos((1/6)*arccos(1 - (27*(1 + x))/16)) + 2*sqrt(6)*sin((1/3)*arcsin((3/4)*sqrt(3/2)*sqrt(1 + x)))).
%F A330793 a(n) ~ 2^(3/2) * 3^(3*n - 1/2) / (sqrt(Pi*n) * 5^(n + 1/2)). - _Vaclav Kotesovec_, Oct 24 2023
%p A330793 a := proc(n) option remember;
%p A330793 if n < 3 then return [1, 2, 8][n+1] fi;
%p A330793 ((60-81*n+27*n^2)*a(n-2) + (22*n^2-28*n+6)*a(n-1))/(5*n*(n-1)) end:
%p A330793 seq(a(n), n=0..24);
%p A330793 # Alternative:
%p A330793 gf := x -> (16 + 8*hypergeom([2/3, 1/3], [1/2], (1+x)*27/32) +
%p A330793 sqrt(18*(1+x))*hypergeom([7/6, 5/6], [3/2], (1+x)*27/32))/48:
%p A330793 ser := series(gf(x), x, 32): evalf(%, 32):
%p A330793 seq(round(coeff(%, x, n)), n=0..24);
%p A330793 # Or:
%p A330793 Gf := x -> (1/(3*sqrt(5 - 27*x)))*(sqrt(5 - 27*x) +
%p A330793 2*sqrt(2)*cos((1/6)*arccos(1 - (27*(1 + x))/16)) +
%p A330793 2*sqrt(6)*sin((1/3)*arcsin((3/4)*sqrt(3/2)*sqrt(1 + x)))):
%p A330793 ser := series(Gf(x), x, 32): evalf(%, 32):
%p A330793 seq(round(coeff(%,x,n)), n=0..24);
%t A330793 a[n_]:= a[n]= If[n<3, 2^n*n!, (2*(n-1)*(11*n-3)*a[n-1] +3*(3*n-4)*(3*n -5)*a[n-2])/(5*n*(n-1))]; (* a=A330793 *)
%t A330793 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Oct 24 2023 *)
%o A330793 (Magma) [1] cat [n le 2 select 2*(3*n-2) else ( 2*(11*n-3)*(n-1)*Self(n-1) + 3*(3*n-4)*(3*n-5)*Self(n-2) )/(5*n*(n-1)): n in [1..30]]; // _G. C. Greubel_, Oct 24 2023
%o A330793 (SageMath)
%o A330793 @CachedFunction
%o A330793 def a(n): # a = A330793
%o A330793     if n<3: return (1,2,8)[n]
%o A330793     else: return (2*(n-1)*(11*n-3)*a(n-1) + 3*(3*n-4)*(3*n-5)*a(n-2))/(5*n*(n-1))
%o A330793 [a(n) for n in range(41)] # _G. C. Greubel_, Oct 24 2023
%Y A330793 Cf. A193737.
%K A330793 nonn
%O A330793 0,2
%A A330793 _Peter Luschny_, Jan 10 2020