A091814 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)!.

1, 11, 203, 5277, 177369, 7324395, 359148915, 20407879485, 1319403598065, 95675323922955, 7692687010986075, 679392106087204125, 65389701973681233225, 6813133489461071047275, 764091914039696003436675

Offset

0,2

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, Mar 15 2004

Links

G. C. Greubel, Table of n, a(n) for n = 0..330

S. Kurz and V. Mishkin, Sets avoiding integral distances, arXiv:1204.0403 [math.MG], 2012. - From N. J. A. Sloane, Oct 03 2012

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, Sep 30 2006

Asymptotics: a(n) ~ 2^(n-2)*((2n)!/n!)*sqrt(Pi/n). - Sascha Kurz, Feb 02 2012

Example

I(3)=5277/13440, a(3) = 5277, b(3)=13440.

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}] (* Robert G. Wilson v, Mar 15 2004 *)

Crossrefs

Sequence in context: A213531 A064748 A191556 * A280957 A020518 A371387

Adjacent sequences: A091811 A091812 A091813 * A091815 A091816 A091817

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)excite.com), Mar 07 2004

Extensions

More terms from Robert G. Wilson v, Mar 15 2004

Status

approved