OFFSET
1,1
FORMULA
G.f.: (-6*x^2+sqrt(1-4*x)*(4*x+1)+6*x-1)/(2*sqrt(1-4*x)).
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
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
From Amiram Eldar, Feb 17 2023: (Start)
Sum_{n>=3} 1/a(n) = 168*Pi^2 - 82262*Pi/(45*sqrt(3)) + 248747/150.
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)
MATHEMATICA
Table[Binomial[2 n - 4, n - 1] (n + 3) / n, {n, 45}] (* Vincenzo Librandi, Apr 15 2016 *)
PROG
(Maxima) taylor((-6*x^2+sqrt(1-4*x)*(4*x+1)+6*x-1)/(2*sqrt(1-4*x)), x, 0, 27);
(Magma) [Binomial(2*n-4, n-1)*(n+3)/n: n in [1..30]]; // Vincenzo Librandi, Apr 15 2016
(PARI) lista(nn) = for(n=1, nn, print1(binomial(2*n-4, n-1)*(n+3)/n, ", ")); \\ Altug Alkan, Apr 15 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Apr 14 2016
STATUS
approved