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%I #40 Jul 11 2024 16:53:35
%S 0,1,1,3,9,29,97,335,1185,4273,15649,58051,217673,823709,3141697,
%T 12065087,46613121,181049025,706549185,2769069091,10894034633,
%U 43008192637,170327978401,676511841743,2694115428769,10755172567089,43032673671137,172538997381155
%N Expansion of -1 - (sqrt(x^4 + 4*x^3 - 2*x^2 - 4*x + 1) - x^2 - 2*x - 1)/(4*x).
%F a(n) = Sum_{k=1..n} (binomial(n-k,k-1)*Sum_{i=0..n-k} 2^i*binomial(k,n-k-i)* binomial(k+i-1,k-1)*(-1)^(n-k-i)/k).
%F D-finite with recurrence: (n+1)*a(n) +2*(-2*n+1)*a(n-1) +2*(-n+2)*a(n-2) +2*(2*n-7)*a(n-3) +(n-5)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020
%F From _Emanuele Munarini_, Jul 11 2024: (Start)
%F a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n+k-1,3*k)*2^k*Catalan(k).
%F a(n) ~ (1/2)*sqrt(sqrt(5)/(2*Pi))*(2+sqrt(5))^n/n^(3/2). (End)
%F a(n) = hypergeom([1/2, 1/2 - n/2, 1 - n/2, n], [1/3, 2/3, 2], 32/27) for n > 0. -_Peter Luschny_, Jul 11 2024
%t CoefficientList[Series[-1-(Sqrt[x^4+4*x^3-2*x^2-4*x+1]-x^2-2*x-1)/(4*x),{x,0,20}],x] (* _Vaclav Kotesovec_, Nov 23 2014 *)
%t a[0] := 0;
%t a[n_] := HypergeometricPFQ[{1/2, (1 - n)/2, 1 - n/2, n}, {1/3, 2/3, 2}, 32/27];
%t Table[a[n], {n, 0, 26}] (* _Peter Luschny_, Jul 11 2024 *)
%o (Maxima) a(n):=sum((binomial(n-k,k-1)*sum(2^i*binomial(k,n-k-i)*binomial(k+i-1,k-1)*(-1)^(n-k-i),i,0,n-k))/k,k,1,n);
%Y Cf. A000108.
%K nonn
%O 0,4
%A _Vladimir Kruchinin_, Nov 22 2014
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