%I #19 Mar 03 2024 09:50:26

%S 1,1,2,5,15,45,142,464,1556,5327,18532,65326,232826,837589,3037472,

%T 11092143,40753626,150541422,558762382,2082871613,7794301294,

%U 29269317708,110263451242,416595676681,1578183767068,5993326380378,22812048907856,87010994947971,332531385362972

%N G.f. satisfies A(x) = (1 + x^4) * (1 + x*A(x)^2).

%F G.f.: A(x) = 2*(1 + x^4) / (1 + sqrt(1-4*x*(1 + x^4)^2)).

%F a(n) = Sum_{k=0..floor(n/4)} binomial(2*n-8*k+1,k) * binomial(2*n-8*k+1,n-4*k) / (2*n-8*k+1).

%F D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-4) +6*(-2*n+9)*a(n-5) +6*(-2*n+17)*a(n-9) +2*(-2*n+25)*a(n-13)=0. - _R. J. Mathar_, Jul 25 2023

%p A364330 := proc(n)

%p add( binomial(2*n-8*k+1,k) * binomial(2*n-8*k+1,n-4*k)/(2*n-8*k+1),k=0..n/4) ;

%p end proc:

%p seq(A364330(n),n=0..80); # _R. J. Mathar_, Jul 25 2023

%t nmax = 28; A[_] = 1;

%t Do[A[x_] = (1 + x^4)*(1 + x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax+1}];

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Mar 03 2024 *)

%o (PARI) a(n) = sum(k=0, n\4, binomial(2*n-8*k+1, k)*binomial(2*n-8*k+1, n-4*k)/(2*n-8*k+1));

%Y Cf. A073157, A215576, A364329.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jul 18 2023