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%I #52 Dec 15 2024 07:23:49
%S 1,1,1,1,1,1,1,467807924713440738696537864469,
%T 17708695183056190642497315530628422295569865119,
%U 8096799621940897567828686854312535486311061114550605367511653,2051563935160591194337436768610392837217226815379395891838337765936509
%N Numerator of n-th Borwein integral divided by Pi/2.
%C The n-th Borwein integral is usually defined as J_n = Integral_{x=-oo..oo} (Product_{k=0..n} sinc(x/(2k+1))) dx.
%C J_n is a rational multiple of Pi/2: J_n/(Pi/2) = a(n) / A144616(n).
%C Alternatively, Weisstein defines "Borwein integral of order 2n+1", the rational number I_{2n+1} = (1/Pi) * Integral_{x=-oo..oo} (Product_{k=0..n} sin(x/(2k+1))/x) dx = J_n / Pi / (2n+1)!!. I_{2n+1} apparently also has numerator a(n), and the denominator is given by A068215(n).
%H Robert G. Wilson v, <a href="/A068214/b068214.txt">Table of n, a(n) for n = 0..12</a>
%H J. M. Borwein, <a href="http://carmamaths.org/resources/jon/hhm.pdf">The Life of Modern Homo Habilis Mathematicus: Experimental Computation and Visual Theorems</a>, 2014; Chapter prepared for John Monaghan, Luc Troche and Jonathan Borwein, "Tools and mathematics: Instruments for learning", Spring-Verlag, 2015.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BorweinIntegrals.html">Borwein Integrals</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Borwein_integral">Borwein integral</a> (From _N. J. A. Sloane_, Feb 25 2012)
%e For n = 0, 1, 2..., the sequence of rational numbers J_n/(Pi/2) is given by 1, 1, 1, 1, 1, 1, 1, 467807924713440738696537864469 / 467807924720320453655260875000 = 1 - 491^7 / (2^3 3^12 5^6 7^7 11^6 13^6)...
%t Table[2/Pi*Integrate[Product[Sinc[x/k], {k, 1, 2*n - 1, 2}], {x, 0, Infinity}], {n, 9}] // Numerator (* _Bill Gosper_, Jan 07 2009 *)
%t borwein[n_] := (2n+1)/4^n Binomial[2n,n] Sum[With[{bg=1+g.(1/(2Range@n+1))}, Times@@g bg^n Sign[bg]], {g,Tuples[{1,-1},n]}];
%t Numerator@Table[borwein[n], {n,0,12}] (* _Andrey Zabolotskiy_, Nov 03 2024 *)
%Y Cf. A068215, A144616 (denominators).
%K nonn,frac,nice
%O 0,8
%A _Eric W. Weisstein_, Feb 21 2002
%E Definition and comments edited by _Andrey Zabolotskiy_, Dec 14 2024, based on contributions from _Bill Gosper_, Jan 07 2009, and _Robert B Fowler_, Oct 28 2024