A068215 -id:A068215 - cp's OEIS frontend

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

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

Offset: 0

Eric W. Weisstein, Feb 21 2002

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).

			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)...

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)

Cf. A068215, A144616 (denominators).

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 *)

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

A221208 Decimal expansion of the Borwein integral with 8 sinc functions.

Original entry on oeis.org

1, 5, 7, 0, 7, 9, 6, 3, 2, 6, 7, 7, 1, 7, 9, 6, 0, 4, 6, 5, 0, 5, 8, 4, 0, 8, 9, 4, 2, 4, 6, 4, 9, 5, 8, 5, 4, 7, 5, 0, 6, 5, 9, 3, 1, 8, 3, 8, 7, 5, 3, 2, 5, 9, 5, 9, 8, 0, 2, 2, 7, 5, 8, 2, 3, 5, 4, 7, 7, 6, 9, 6, 2, 7, 6, 6, 9, 2, 6, 3, 9, 1, 0, 7, 0, 4, 9, 6, 6, 6, 1, 7, 9, 3, 8, 6, 3, 4, 7, 3, 4, 0, 5, 0, 3

Offset: 1

Jean-François Alcover, Feb 21 2013

The difference from Pi/2 (A019669) is approximately 0.231006*10^(-10).

			1.57079632677179604650584089424649585475065931838753259598...

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Borwein Integrals
Wikipedia, Borwein integral

Cf. A000796, A019669, A068214, A068215, A144616, A091473.

Integrate[Product[Sinc[x/(2*k+1)], {k, 0, 7}], {x, 0, Infinity}] // RealDigits[#, 10, 105]& // First

Equals 467807924713440738696537864469/935615849440640907310521750000*Pi. - Alois P. Heinz, Feb 28 2020

Equals Pi/2*A068214(7)/A144616(7). - Andrey Zabolotskiy, Jan 04 2023

A280841 Numerator of Integral_{x>=0} Product_{k=1..n} Sinc(x/k) dx / Pi.

Original entry on oeis.org

1, 1, 1, 1727, 20652479, 2059268143, 24860948333867803, 145905074443586569379, 4567419249415312673370820607, 1642142815363470261591271553081, 4093745592094627817260334517735412136353665283

Offset: 1

Seiichi Manyama, Jan 08 2017

Let I(n) be defined by I(n) = Integral_{x>=0} Product_{k=1..n} Sinc(x/k) dx.

I(1) = I(2) = I(3) = Pi/2, however I(4) = Pi/2 - Pi/3456.

			I(4) = 1727*Pi/3456. So a(4) = 1727.
I(5) = 20652479*Pi/41472000. So a(5) = 20652479.
I(6) = 2059268143*Pi/4147200000. So a(6) = 2059268143.
I(7) = 24860948333867803*Pi/50185433088000000. So a(7) = 24860948333867803.

Robert G. Wilson v, Table of n, a(n) for n = 1..15

Cf. A068214, A068215, A280842 (denominators).

f[n_] := Numerator[Integrate[Product[Sinc[x/k], {k, n}], {x, 0, Infinity}]/Pi]; Array[f, 11] (* Robert G. Wilson v, Jan 29 2017 *)

a(8)-a(11) from Alois P. Heinz, Jan 09 2017

A280842 Denominator of Integral_{x>=0} Product_{k=1..n} Sinc(x/k) dx / Pi.

Original entry on oeis.org

2, 2, 2, 3456, 41472000, 4147200000, 50185433088000000, 295090346557440000000, 9251918060437194670080000000, 3330690501757390081228800000000, 8312243866372850396258184884618526720000000000

Offset: 1

Seiichi Manyama, Jan 08 2017

Let I(n) be defined by I(n) = Integral_{x>=0} Product_{k=1..n} Sinc(x/k) dx. I(1) = I(2) = I(3) = Pi/2, however I(4) = Pi/2 - Pi/3456.

			I(4) = 1727*Pi/3456. So a(4) = 3456.
I(5) = 20652479*Pi/41472000. So a(5) = 41472000.
I(6) = 2059268143*Pi/4147200000. So a(6) = 4147200000.
I(7) = 24860948333867803*Pi/50185433088000000. So a(7) = 50185433088000000.

Robert G. Wilson v, Table of n, a(n) for n = 1..15

Cf. A068214, A068215, A280841 (numerators).

f[n_] := Denominator[Integrate[Product[Sinc[x/k], {k, n}], {x, 0, Infinity}]/Pi]; Array[f, 11] (* Robert G. Wilson v, Jan 29 2017 *)

a(8)-a(11) from Alois P. Heinz, Jan 10 2017

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

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A221208 Decimal expansion of the Borwein integral with 8 sinc functions.

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A280841 Numerator of Integral_{x>=0} Product_{k=1..n} Sinc(x/k) dx / Pi.

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A280842 Denominator of Integral_{x>=0} Product_{k=1..n} Sinc(x/k) dx / Pi.

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Mathematica

Extensions