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

2, 7, 24, 83, 292, 1046, 3808, 14051, 52412, 197202, 747120, 2846318, 10892936, 41844172, 161247104, 623034403, 2412871916, 9363311482, 36399254864, 141721774138, 552572485496, 2157194452852, 8431104269504, 32986010380558, 129177323979992, 506313914434036, 1986097541692128

Offset: 1

Author

Al Hakanson (hawkuu(AT)excite.com), Nov 24 2003

Keywords

• nonn

Examples

			a(3) = 24.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A082590.

Programs

Maple

A088854 := n -> 2^(n-1)*JacobiP(n, 1/2, -1 - n, 3):
seq(simplify(A088854(n)), n = 1..26);  # Peter Luschny, Jan 22 2025

Mathematica

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 *)
Table[2^(n-1)+Sum[2^(n-k)*Binomial[2*k-1,k], {k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Oct 28 2012 *)

PARI

x='x+O('x^66); Vec(-1/2+1/(2*(1-2*x)*sqrt(1-4*x))) \\ Joerg Arndt, May 10 2013

Formula

G.f.: -1/2 + 1/(2*(1-2*x)*sqrt(1-4*x)). - Vladeta Jovovic, Dec 14 2003

Recurrence: n*a(n) = 2*(3*n-1)*a(n-1) - 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012

a(n) ~ 4^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012

a(n) = 2^(n-1) + Sum_{k=1..n} 2^(n-k)*C(2*k-1,k). - Vaclav Kotesovec, Oct 28 2012

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

a(n) = 2^(n-1)*JacobiP(n, 1/2, -1 - n, 3). - Peter Luschny, Jan 22 2025

Extensions

More terms from Robert G. Wilson v, Feb 27 2004