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Evaluating a simple integral
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Peter Luschny
Apr 5, 2012, 11:49:08 AM4/5/12
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Hi all!
A := n -> (1/Pi)*int((sin(x)/
x)^(2*n)*sin(2*n*x)*tan(x),x=0..infinity);
seq(A(i),i=1..6);
1/2, -1/6, 1/15, -17/630, 31/2835, -691/155925
Thereafter my very old Maple V fails. How does Maple 16 continue?
Peter
A := n -> (1/Pi)*int((sin(x)/
x)^(2*n)*sin(2*n*x)*tan(x),x=0..infinity);
seq(A(i),i=1..6);
1/2, -1/6, 1/15, -17/630, 31/2835, -691/155925
Thereafter my very old Maple V fails. How does Maple 16 continue?
Peter
Axel Vogt
Apr 5, 2012, 12:48:29 PM4/5/12
to
10922/6081075, -929569/1277025750, 3202291/10854718875,
-221930581/1856156927625, 9444233042/194896477400625,
-56963745931/2900518163668125
Gruß
Peter Luschny
Apr 5, 2012, 1:31:55 PM4/5/12
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By the way, any idea what these numbers are?
Axel Vogt
Apr 5, 2012, 4:03:43 PM4/5/12
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Guess you want to check OEIS.org and miss results for that.
For seq(A(i),i=1..18) I almost get https://oeis.org/A046990 with
sign changes for the numerator, but nothing (directly) for the
denominator.
[1/2, -1/6, 1/15, -17/630, 31/2835, -691/155925, 10922/6081075,
-929569/1277025750, 3202291/10854718875, -221930581/1856156927625,
9444233042/194896477400625, -56963745931/2900518163668125,
29435334228302/3698160658676859375,
-4187321758505342/1298054391195577640625,
344502690252804724/263505041412702261046875,
-129848163681107301953/245059688513813102773593750,
868320396104950823611/4043484860477916195764296875,
-209390615747646519456961/2405873491984360136479756640625]
Peter Luschny
Apr 5, 2012, 4:37:03 PM4/5/12
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On 5 Apr., 22:03, Axel Vogt <&nore...@axelvogt.de> wrote:
> For seq(A(i),i=1..18) I almost gethttps://oeis.org/A046990with
W := proc(n) (4^n*(4^n-1)/2)*bernoulli(2*n)/(2*n)! end;
seq((W(i)),i=1..18);
which agrees with your data.
> For seq(A(i),i=1..18) I almost gethttps://oeis.org/A046990with
> sign changes for the numerator, but nothing (directly) for the
> denominator.
>
> [1/2, -1/6, 1/15, -17/630, 31/2835, -691/155925, 10922/6081075,
> -929569/1277025750, 3202291/10854718875, -221930581/1856156927625,
> 9444233042/194896477400625, -56963745931/2900518163668125,
> 29435334228302/3698160658676859375,
> -4187321758505342/1298054391195577640625,
> 344502690252804724/263505041412702261046875,
> -129848163681107301953/245059688513813102773593750,
> 868320396104950823611/4043484860477916195764296875,
> -209390615747646519456961/2405873491984360136479756640625]
Thank you Axel! In the meantime I found out:
> denominator.
>
> [1/2, -1/6, 1/15, -17/630, 31/2835, -691/155925, 10922/6081075,
> -929569/1277025750, 3202291/10854718875, -221930581/1856156927625,
> 9444233042/194896477400625, -56963745931/2900518163668125,
> 29435334228302/3698160658676859375,
> -4187321758505342/1298054391195577640625,
> 344502690252804724/263505041412702261046875,
> -129848163681107301953/245059688513813102773593750,
> 868320396104950823611/4043484860477916195764296875,
> -209390615747646519456961/2405873491984360136479756640625]
W := proc(n) (4^n*(4^n-1)/2)*bernoulli(2*n)/(2*n)! end;
seq((W(i)),i=1..18);
which agrees with your data.
Peter Luschny
Apr 6, 2012, 7:20:25 PM4/6/12
to
> how the integrals came up?
Well, a guy named William Rowan Hamilton got interested in them.
> Guess you want to check OEIS.org and miss results for that.
Good guess :-) Yes, I wanted to check if the quaternions confused
William.
Here is the rest of the story: http://oeis.org/A181993
Thanks again for your help!
Well, a guy named William Rowan Hamilton got interested in them.
> Guess you want to check OEIS.org and miss results for that.
William.
Here is the rest of the story: http://oeis.org/A181993
Thanks again for your help!
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