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A088854 a(n) = (2^(n-1))*(integral_{x=0 to 1} (1+x^2)^n dx)/(integral_{x=0 to 1} (1-x^2)^n dx). 1
2, 7, 24, 83, 292, 1046, 3808, 14051, 52412, 197202, 747120, 2846318, 10892936, 41844172, 161247104, 623034403, 2412871916, 9363311482, 36399254864, 141721774138, 552572485496, 2157194452852, 8431104269504, 32986010380558 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

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

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

EXAMPLE

a(3) = 24.

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 *)

PROG

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

CROSSREFS

Cf. A082590.

Sequence in context: A021000 A003480 A020727 * A000777 A144170 A297345

Adjacent sequences:  A088851 A088852 A088853 * A088855 A088856 A088857

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified April 25 04:26 EDT 2019. Contains 322451 sequences. (Running on oeis4.)