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%I #20 Aug 29 2023 05:22:36
%S 1,1,2,5,15,49,168,595,2160,7997,30083,114660,441840,1718531,6737820,
%T 26600784,105659970,421949492,1693120779,6823018035,27602090087,
%U 112053680381,456343848121,1863893501065,7633232165286,31337360839387,128944120202510
%N G.f. satisfies A(x) = 1 + x*A(x)^2/(1 - x^3*A(x)^3).
%F a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(2*n-3*k+1,n-3*k)/(2*n-3*k+1).
%F D-finite with recurrence 31*n*(626109182191*n-1858292669035) *(n-1)*(n+1) *a(n) -n*(n-1) *(244150473843619*n^2 -1454194662255591*n +2175006457069082) *a(n-1) +3*(n-1) *(292927551362415*n^3 -2593205532882651*n^2 +7084566217454162*n -5823331737745632)*a(n-2) +(-843955616916167*n^4 +9932491073296715*n^3 -42016891739306929*n^2 +76184884157722453*n -50166914106142776) *a(n-3) +18*(1509721335071*n^4 -40413442328880*n^3 +330301781039401*n^2 -1078322794857576*n +1231650372542192) *a(n-4) +18*(39673125909769*n^4 -598320530478001*n^3 +3228489073613917*n^2 -7321259523567459*n +5788776339353646) *a(n-5) +27*(n-5) *(3102413205331*n^3 -35996479327373*n^2 +114122791959960*n -64735736097804) *a(n-6) -243*(n-6) *(n-7)*(475638134099*n^2 -2399948859181*n +2877042451214) *a(n-7) -243*(45857481910*n -35520400961) *(n-5) *(n-7) *(n-8)*a(n-8)=0. - _R. J. Mathar_, Aug 29 2023
%p A364833 := proc(n)
%p add( binomial(n-2*k-1,k)*binomial(2*n-3*k+1,n-3*k)/ (2*n-3*k+1),k=0..floor(n/3)) ;
%p end proc:
%p seq(A364833(n),n=0..80); # _R. J. Mathar_, Aug 29 2023
%o (PARI) a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(2*n-3*k+1, n-3*k)/(2*n-3*k+1));
%Y Cf. A112806, A219537.
%Y Cf. A218251, A364161, A365247.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 28 2023
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