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 1-4 of 4 results.

A300707 Decimal expansion of Pi^4/96.

Original entry on oeis.org

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

Views

Author

Iaroslav V. Blagouchine, Mar 11 2018

Keywords

Comments

Also the sum of the series Sum_{n>=0} (1/(2n+1)^4), whose value is obtained from zeta(4) given by L. Euler in 1735: Sum_{n>=0} (2n+1)^(-s) = (1-2^(-s))*zeta(s).
For the partial sums of this series see A120269/A128493. - Wolfdieter Lang, Sep 02 2019

Examples

			1.0146780316041920545462534655073449088513290174238064...

Links

Crossrefs

Cf. A013662, A092425, A111003, A120269, A128493, A300709, A300710, A300731.

Programs

  • MATLAB
    format long; pi^4/96
  • Maple
    evalf((1/96)*Pi^4, 120)
  • Mathematica
    RealDigits[Pi^4/96, 10, 120][[1]]
  • PARI
    default(realprecision, 120); Pi^4/96

Formula

Equals A092425/96. - Omar E. Pol, Mar 11 2018
Equals (15/16)*zeta(4) = (15/16)*A013662. - Wolfdieter Lang, Sep 02 2019
Equals Sum_{k>=1} 1/(2*k-1)^4. - Sean A. Irvine, Mar 25 2025
Equals lambda(4), where lambda is the Dirichlet lambda function. - Michel Marcus, Aug 15 2025

A128493 Denominators of partial sums for a series for (Pi^4)/96.

Original entry on oeis.org

1, 81, 50625, 121550625, 9845600625, 144149438750625, 4117052120156600625, 4117052120156600625, 343860310127599440800625, 44812219476138886724578250625, 44812219476138886724578250625
Offset: 1

Views

Author

Wolfdieter Lang Apr 04 2007

Keywords

Comments

Numerators are given in A120269.
See the comments and the W. Lang link under A120269.

Programs

  • Mathematica
    a[n_] := (Pi^4 - PolyGamma[3, n + 1/2])/96 // Simplify // Denominator; Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Dec 05 2013 *)

Formula

a(n)=denominator(r(n)) with r(n):=sum(1/(2*k-1)^4,k=1..n).

A164655 Numerators of partial sums of Theta(3) = Sum_{j>=1} 1/(2*j-1)^3.

Original entry on oeis.org

1, 28, 3527, 1213136, 32797547, 43684790932, 96017087247229, 96044168328256, 471956397645187853, 3237597973008257555852, 462561506842656976961, 5628425850334528955928112, 703596058798919360293439483, 18998011529681231695738912916, 463360571051954739540899597748949
Offset: 1

Views

Author

Wolfdieter Lang, Oct 16 2009

Keywords

Comments

Warning: Usually, Theta3(x) = Sum_{n=-oo..+oo} x^(n^2). - Joerg Arndt, Mar 31 2024
The denominators look like those given for the partial sums of another series in A128507.
Rationals (partial sums) Theta(3,n) := Sum_{j=1..n} 1/(2*j-1)^3 (in lowest terms). The limit of these rationals is Theta(3) = (1-1/2^3)*Zeta(3) approximately 1.051799790 (Zeta(n) is the Euler-Riemann zeta function).
This is a member of the k-family of rational sequences Theta(k,n) := Sum_{j=1..n} 1/(2*j-1)^k, k >= 1, which coincides for k=1 with A025550/A025547 (but only for the first 38 terms), for k=2 with A120268/A128492, for k=3 with a(n)/A128507(n) (the denominators may depart for higher n values), A120269/A128493 and A164656/A164657, for k=4 and 5, respectively.

Examples

			Rationals Theta(3,n): [1, 28/27, 3527/3375, 1213136/1157625, 32797547/31255875, 43684790932/41601569625, ...].

Links

Programs

  • Mathematica
    r[n_] := Sum[1/(2*j-1)^3, {j, 1, n}]; (* or r[n_] := (PolyGamma[2, n+1/2] - PolyGamma[2, 1/2])/16 // FullSimplify; *) Table[r[n] // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *)

Formula

a(n) = numerator(Theta(3,n)) = numerator(Sum_{j=1..n} 1/(2*j-1)^3), n >= 1.
Theta(3,n) = (-Psi(2, 1/2) + Psi(2, n+1/2))/16, n >= 1, where Psi(n, k) = Polygamma(n,k) is the n-th derivative of the digamma function. Psi(2, 1/2) = -14*Zeta(3). - Jean-François Alcover, Dec 02 2013

A164656 Numerators of partial sums of Theta(5) = sum( 1/(2*j-1)^5, j=1..infinity ).

Original entry on oeis.org

1, 244, 762743, 12820180976, 3115356499043, 501734380891571068, 186290962962179367466549, 186291207179611798681792, 264507060005034822095008296869, 654945930087597102815813733559637156, 654946089730308117005814730177159031, 4215458332009996232497953858159263996273008
Offset: 1

Views

Author

Wolfdieter Lang, Oct 16 2009

Keywords

Comments

The denominators are given by A164657.
Rationals (partial sums) Theta(5,n) := sum(1/(2*j-1)^5,j=1..n) (in lowest terms). The limit of these rationals is Theta(5)= (1-1/2^5)*Zeta(5) approximately 1.004523763.., see A013663.
This is a member of the k-family of rational sequences Theta(k,n):=sum(1/(2*j-1)^k,j=1..n), k>=1, which includes A025550/A025547 (but only for the first 38 entries), A120268/A128492, A164655(n)/A128507(n) (the denominators may depart for higher n values), A120269/A128493, a(n)/A164657, for k=1..5.

Examples

			Rationals Theta(5,n): [1, 244/243, 762743/759375, 12820180976/12762815625, 3115356499043/3101364196875,...].

Links

Programs

  • Mathematica
    r[n_] := Sum[1/(2*j-1)^5, {j, 1, n}]; (* or r[n_] := (PolyGamma[4, n+1/2] - PolyGamma[4, 1/2])/768 // FullSimplify; *) Table[r[n] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 02 2013 *)

Formula

a(n) = numer(Theta(5,n))= numerator(sum(1/(2*j-1)^5,j=1..n)), n>=1.
Theta(5,n) = (-Psi(4, 1/2) + Psi(4, n+1/2))/(4!*2^5), n >= 1, with Psi(n,k) = Polygamma(n,k) is the n^th derivative of the digamma function. Psi(4, 1/2) = -4!*31*Zeta(5). - Jean-François Alcover, Dec 02 2013
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.