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A348618 a(n) = (1+(-1)^n)/2*4^n*(C((3*n)/2-1,n))+(1-(-1)^n)/2*((C((3*n-1)/2,n))*(C(3*n-1,(3*n-1)/2)))/(C(n-1,(n-1)/2)).

%I A348618 #20 Mar 06 2022 08:29:38
%S A348618 1,2,16,140,1280,12012,114688,1108536,10813440,106234700,1049624576,
%T A348618 10418726760,103817412608,1037865473400,10404558274560,
%U A348618 104557533120240,1052941297385472,10623352887172620,107358720517734400,1086563988284497800,11011614449734778880
%N A348618 a(n) = (1+(-1)^n)/2*4^n*(C((3*n)/2-1,n))+(1-(-1)^n)/2*((C((3*n-1)/2,n))*(C(3*n-1,(3*n-1)/2)))/(C(n-1,(n-1)/2)).
%F A348618 G.f.: (288*x^2*cos(arcsin(216*x^2-1)/3))/(sqrt(432*x^2-46656*x^4)*(2*sin(arcsin(216*x^2-1)/3)+1)).
%F A348618 Conjecture: D-finite with recurrence n*(n-1)*a(n) -12*(3*n-2)*(3*n-4)*a(n-2)=0. - _R. J. Mathar_, Mar 06 2022
%p A348618 a:= n-> ceil(4^n*binomial(3*n/2, n)/3):
%p A348618 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 25 2021
%t A348618 a[n_] := If[EvenQ[n], 4^n * Binomial[3*n/2 - 1, n], Binomial[(3*n - 1)/2, n] * Binomial[3*n - 1, (3*n - 1)/2] / Binomial[n - 1, (n - 1)/2]]; Array[a, 18, 0] (* _Amiram Eldar_, Oct 25 2021 *)
%o A348618 (Maxima)
%o A348618 a(n):=if evenp(n) then 4^n*binomial(3*n/2-1,n) else ((binomial((3*n-1)/2,n))*
%o A348618     (binomial(3*n-1,(3*n-1)/2)))/binomial(n-1,(n-1)/2);
%Y A348618 Cf. A244038.
%K A348618 nonn
%O A348618 0,2
%A A348618 _Vladimir Kruchinin_, Oct 25 2021