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A091814
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Numerator of I(n)=(integral_{x=0..1/2}(1-x^2)^n,dx), where the denominator is b(n)=2^n*(2*n+2)!/(n+1)!.
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1
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1, 11, 203, 5277, 177369, 7324395, 359148915, 20407879485, 1319403598065, 95675323922955, 7692687010986075, 679392106087204125, 65389701973681233225, 6813133489461071047275, 764091914039696003436675
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also numerator of a(n)=(integral_{x=0 to 1}(4-x^2)^n dx)m where the denominator is b(n)=(2*n+2)!/((n+1)!*2^(n+1)). E.g. a(3)=5277/105. b(3)=105.
Also numerator of I(n)=(integral_{x=0 to 1}(4-x^2)^n dx) where the denominator is b(n)=(2*n+2)!/((n+1)!*2^(n+1)). E.g. I(3)=5277/105. b(3)=105. - Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 15 2004
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FORMULA
| a(n)=2^(n-1)*((2n)!/n!)*J(n) where J(n)=integral(t=0,Pi/6,cos(t)^(2n-1)dt) is given by the order 2 recursion : J(1)=1/2, J(2)=11/24, J(n)=1/(8*n-4)*((14*n-17)*J(n-1)-6*(n-2)*J(n-1)) - Benoit Cloitre (abmt(AT)orange.fr), Sep 30 2006
Asymptotics: a(n) ~ 2^(n-2)*((2n)!/n!)*sqrt(pi/n). [Sascha Kurz, Feb 02 2012]
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EXAMPLE
| I(3)=5277/13440, a(3) = 5277, b(3)=13440.
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MATHEMATICA
| A091814[n_] := Integrate[(1 - x^2)^n, {x, 0, 1/2}]2^n*(2*n + 2)!/(n + 1)!; Table[ A091814[n], {n, 0, 14}] (from Robert G. Wilson v Mar 15 2004)
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CROSSREFS
| Sequence in context: A088295 A064748 A191556 * A020518 A196849 A196699
Adjacent sequences: A091811 A091812 A091813 * A091815 A091816 A091817
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KEYWORD
| nonn,changed
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AUTHOR
| Al Hakanson (hawkuu(AT)excite.com), Mar 07 2004
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 15 2004
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