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A088854 a(n) = (2^(n-1))*(Integral_{x=0..1} (1+x^2)^n dx)/(Integral_{x=0..1} (1-x^2)^n dx).

%I A088854 #30 Jan 22 2025 08:52:33
%S A088854 2,7,24,83,292,1046,3808,14051,52412,197202,747120,2846318,10892936,
%T A088854 41844172,161247104,623034403,2412871916,9363311482,36399254864,
%U A088854 141721774138,552572485496,2157194452852,8431104269504,32986010380558,129177323979992,506313914434036,1986097541692128
%N A088854 a(n) = (2^(n-1))*(Integral_{x=0..1} (1+x^2)^n dx)/(Integral_{x=0..1} (1-x^2)^n dx).
%H A088854 Vincenzo Librandi, <a href="/A088854/b088854.txt">Table of n, a(n) for n = 1..200</a>
%F A088854 G.f.: -1/2 + 1/(2*(1-2*x)*sqrt(1-4*x)). - _Vladeta Jovovic_, Dec 14 2003
%F A088854 Recurrence: n*a(n) = 2*(3*n-1)*a(n-1) - 4*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 14 2012
%F A088854 a(n) ~ 4^n/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 14 2012
%F A088854 a(n) = 2^(n-1) + Sum_{k=1..n} 2^(n-k)*C(2*k-1,k). - _Vaclav Kotesovec_, Oct 28 2012
%F A088854 2*a(n) = Sum_{k=0..n} C(2k,k)*C(2(n-k),n-k)/C(n,k). - _Zhi-Wei Sun_, Oct 14 2019
%F A088854 a(n) = 2^(n-1)*JacobiP(n, 1/2, -1 - n, 3). - _Peter Luschny_, Jan 22 2025
%e A088854 a(3) = 24.
%p A088854 A088854 := n -> 2^(n-1)*JacobiP(n, 1/2, -1 - n, 3):
%p A088854 seq(simplify(A088854(n)), n = 1..26);  # _Peter Luschny_, Jan 22 2025
%t A088854 f[n_] := 2^(n - 1)Integrate[(1 + x^2)^n, {x, 0, 1}] / Integrate[(1 - x^2)^n, {x, 0, 1}]; Table[ f[n], {n, 1, 24}] (* _Robert G. Wilson v_, Feb 27 2004 *)
%t A088854 Table[2^(n-1)+Sum[2^(n-k)*Binomial[2*k-1,k], {k,1,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 28 2012 *)
%o A088854 (PARI) x='x+O('x^66); Vec(-1/2+1/(2*(1-2*x)*sqrt(1-4*x))) \\ _Joerg Arndt_, May 10 2013
%Y A088854 Cf. A082590.
%K A088854 nonn
%O A088854 1,1
%A A088854 Al Hakanson (hawkuu(AT)excite.com), Nov 24 2003
%E A088854 More terms from _Robert G. Wilson v_, Feb 27 2004