A068214 -id:A068214 - cp's OEIS frontend

A068215 Denominator of Borwein integral of order 2n+1, as defined by Weisstein.

2, 6, 30, 210, 1890, 20790, 270270, 1896516717212415135141110350293750000, 1220462921565155916674902677397230198502690752000000000

Offset: 0

Eric W. Weisstein, Feb 21 2002

Eric Weisstein's World of Mathematics, Borwein Integrals
Wikipedia, Borwein integral [From _N. J. A. Sloane_, Feb 25 2012]

Cf. A068214 (supposed numerators), A144616 (denominators of the conventional Borwein integrals).

i[n_] := Times@@(Sin[x/# ]&/@Range[1, n, 2])/x^((n+1)/2)/Pi; Denominator[Table[Integrate[i[n], {x, 0, \[Infinity]}], {n, 1, 19, 2}]]

a(n) = A144616(n+1)*A097801(n+1) [assuming that the numerators are really A068214]. - Andrey Zabolotskiy, Oct 18 2016

Name edited by Andrey Zabolotskiy, Dec 14 2024

A144616 Denominator of n-th Borwein integral divided by Pi/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 467807924720320453655260875000, 17708695394150597647449176493763755467520000000, 8096800377970649960875919032857634716820075076062381575000000, 2051564503724359411435325207087513361930253427318374450656960000000000

Offset: 0

N. J. A. Sloane, Jan 15 2009, based on email from Bill Gosper

See comments in A068214.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..20 (terms 0..12 from _Robert G. Wilson v_)

Cf. A068214 (numerators).

(Mma 7.0) Table[2/Pi*Integrate[Product[Sinc[x/k], {k, 1, 2*n + 1, 2}], {x, 0, Infinity }], {n, 0, 7}]

a(9)-a(11) from Robert G. Wilson v, Nov 04 2013

New name, offset changed from 1 to 0 by Andrey Zabolotskiy, Dec 14 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

cp's OEIS Frontend

A068215 Denominator of Borwein integral of order 2n+1, as defined by Weisstein.

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A144616 Denominator 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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