%I #15 Oct 25 2023 09:28:36

%S 1,1,1,2,6,17,48,144,449,1422,4568,14893,49139,163665,549570,1858754,

%T 6326343,21651064,74462327,257219221,892047965,3104749126,10841192392,

%U 37967942203,133333407639,469405472729,1656383420850,5857371543403,20754268304707

%N G.f. satisfies A(x) = 1 + x*A(x)/(1 - x^2*A(x)^3).

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

%F D-finite with recurrence -9*n*(3*n-5) *(3*n+2) *(15657757169*n -38967750523)*a(n) +3*(1246945698477*n^4 -4744568003544*n^3 +3294337649527*n^2 +2214578323972*n -1078893934272) *a(n-1) +6*(98125454565*n^4 -4049050969593*n^3 +21710764341344*n^2 -39026642938410*n +22772957131188) *a(n-2) +6*(1426531749264*n^4 -6603349282173*n^3 -4098111856085*n^2 +51689999346882*n -56245738276010) *a(n-3) +6*(2322713957130*n^4 -32736762801117*n^3 +166244031312630*n^2 -356896536324983*n +268070043432100) *a(n-4) -6*(n-5) *(2*n-9) *(613164767527*n^2 -4657829502565*n +8148618486058) *a(n-5) +2*(n-6) *(2*n-11) *(271184324539*n^2 -2272760427224*n +4256723647917) *a(n-6) -4*(6162243349*n -17166617798) *(2*n-13)*(n-6) *(n-7)*a(n-7)=0. - _R. J. Mathar_, Aug 29 2023

%p A365244 := proc(n)

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

%p end proc:

%p seq(A365244(n),n=0..80); # _R. J. Mathar_, Aug 29 2023

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

%t Do[A[x_] = 1 + x*A[x]/(1 - x^2*A[x]^3) + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 25 2023 *)

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

%Y Cf. A212383, A365245.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Aug 28 2023