A068214 Numerator of n-th Borwein integral divided by Pi/2.

1, 1, 1, 1, 1, 1, 1, 467807924713440738696537864469, 17708695183056190642497315530628422295569865119, 8096799621940897567828686854312535486311061114550605367511653, 2051563935160591194337436768610392837217226815379395891838337765936509

Offset: 0

Author

Eric W. Weisstein, Feb 21 2002

Keywords

• nonn
• frac
• nice

Comments

The n-th Borwein integral is usually defined as J_n = Integral_{x=-oo..oo} (Product_{k=0..n} sinc(x/(2k+1))) dx.

J_n is a rational multiple of Pi/2: J_n/(Pi/2) = a(n) / A144616(n).

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).

Examples

			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)...
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..12
• J. M. Borwein, The Life of Modern Homo Habilis Mathematicus: Experimental Computation and Visual Theorems, 2014; Chapter prepared for John Monaghan, Luc Troche and Jonathan Borwein, "Tools and mathematics: Instruments for learning", Spring-Verlag, 2015.
• Eric Weisstein's World of Mathematics, Borwein Integrals
• Wikipedia, Borwein integral (From _N. J. A. Sloane_, Feb 25 2012)

Crossrefs

Cf. A068215, A144616 (denominators).

Programs

Mathematica

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 *)
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]}];
Numerator@Table[borwein[n], {n,0,12}] (* Andrey Zabolotskiy, Nov 03 2024 *)

Extensions

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