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%I A364330 #19 Mar 03 2024 09:50:26
%S A364330 1,1,2,5,15,45,142,464,1556,5327,18532,65326,232826,837589,3037472,
%T A364330 11092143,40753626,150541422,558762382,2082871613,7794301294,
%U A364330 29269317708,110263451242,416595676681,1578183767068,5993326380378,22812048907856,87010994947971,332531385362972
%N A364330 G.f. satisfies A(x) = (1 + x^4) * (1 + x*A(x)^2).
%F A364330 G.f.: A(x) = 2*(1 + x^4) / (1 + sqrt(1-4*x*(1 + x^4)^2)).
%F A364330 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 A364330 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 A364330 := proc(n)
%p A364330     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 A364330 end proc:
%p A364330 seq(A364330(n),n=0..80); # _R. J. Mathar_, Jul 25 2023
%t A364330 nmax = 28; A[_] = 1;
%t A364330 Do[A[x_] = (1 + x^4)*(1 + x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax+1}];
%t A364330 CoefficientList[A[x], x] (* _Jean-François Alcover_, Mar 03 2024 *)
%o A364330 (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 A364330 Cf. A073157, A215576, A364329.
%K A364330 nonn,easy
%O A364330 0,3
%A A364330 _Seiichi Manyama_, Jul 18 2023

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