cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 467807924713440738696537864469, 17708695183056190642497315530628422295569865119, 8096799621940897567828686854312535486311061114550605367511653, 2051563935160591194337436768610392837217226815379395891838337765936509
Offset: 0

Views

Author

Eric W. Weisstein, Feb 21 2002

Keywords

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

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

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

Original entry on oeis.org

2, 6, 30, 210, 1890, 20790, 270270, 1896516717212415135141110350293750000, 1220462921565155916674902677397230198502690752000000000
Offset: 0

Views

Author

Eric W. Weisstein, Feb 21 2002

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]]

Formula

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

Extensions

Name edited 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

Views

Author

Jean-François Alcover, Feb 21 2013

Keywords

Comments

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

Examples

			1.57079632677179604650584089424649585475065931838753259598...

Links

Crossrefs

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

Programs

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

Formula

Equals 467807924713440738696537864469/935615849440640907310521750000*Pi. - Alois P. Heinz, Feb 28 2020
Equals Pi/2*A068214(7)/A144616(7). - Andrey Zabolotskiy, Jan 04 2023
Showing 1-3 of 3 results.

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