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G.f. A(x) satisfies A(x) = 1 - x/A(x) * (1 - A(x) - A(x)^2).
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%I #18 Apr 22 2024 13:17:08

%S 1,1,2,3,3,1,-2,-1,10,25,12,-65,-151,-7,588,1083,-437,-5247,-7732,

%T 7943,47503,53793,-105312,-430117,-343042,1249801,3866558,1730019,

%U -13996095,-34243895,-1947202,150962375,296101866,-121857183,-1582561868,-2468098041,2529520767

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

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

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

%F G.f.: A(x) = 2*x/(1+x - sqrt(1-2*x+5*x^2)).

%F D-finite with recurrence n*a(n) +3*(-n+1)*a(n-1) +(7*n-18)*a(n-2) +5*(-n+3)*a(n-3)=0. - _R. J. Mathar_, Apr 22 2024

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

%Y Cf. A002212, A108447, A364792, A365193.

%K sign

%O 0,3

%A _Seiichi Manyama_, Apr 11 2024