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%I #23 Feb 17 2023 10:05:00
%S 4,0,2,7,24,84,300,1089,4004,14872,55692,209950,795872,3031032,
%T 11589240,44462565,171085500,660009840,2551955340,9887121090,
%U 38374857840,149184555000,580807904040,2264193450090,8837215647624,34529741789904,135054066707000
%N a(n) = binomial(2*n-4,n-1)*(n+3)/n.
%F G.f.: (-6*x^2+sqrt(1-4*x)*(4*x+1)+6*x-1)/(2*sqrt(1-4*x)).
%F D-finite with recurrence: n*(n+2)*(n-3)*a(n) -2*(n-2)*(2*n-5)*(n+3)*a(n-1)=0. - _R. J. Mathar_, Jun 07 2016
%F D-finite with recurrence: n*a(n) +2*(-5*n+7)*a(n-1) +6*(5*n-14)*a(n-2) +12*(-2*n+9)*a(n-3)=0. - _R. J. Mathar_, Jan 27 2020
%F From _Amiram Eldar_, Feb 17 2023: (Start)
%F Sum_{n>=3} 1/a(n) = 168*Pi^2 - 82262*Pi/(45*sqrt(3)) + 248747/150.
%F Sum_{n>=3} (-1)^(n+1)/a(n) = 65084*log(phi)/(5*sqrt(5)) - 6048*log(phi)^2 - 70019/50, where phi is the golden ratio (A001622). (End)
%t Table[Binomial[2 n - 4, n - 1] (n + 3) / n, {n, 45}] (* _Vincenzo Librandi_, Apr 15 2016 *)
%o (Maxima) taylor((-6*x^2+sqrt(1-4*x)*(4*x+1)+6*x-1)/(2*sqrt(1-4*x)),x,0,27);
%o (Magma) [Binomial(2*n-4,n-1)*(n+3)/n: n in [1..30]]; // _Vincenzo Librandi_, Apr 15 2016
%o (PARI) lista(nn) = for(n=1, nn, print1(binomial(2*n-4, n-1)*(n+3)/n, ", ")); \\ _Altug Alkan_, Apr 15 2016
%Y Cf. A000108, A001622, A271622.
%K nonn
%O 1,1
%A _Vladimir Kruchinin_, Apr 14 2016
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