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.

A130550 Denominators of partial sums for a series for 2*Zeta(2)/3 = (Pi^2)/9.

Original entry on oeis.org

1, 12, 180, 1008, 8400, 118800, 75675600, 302702400, 15437822400, 26665329600, 3226504881600, 5708431713600, 964724959598400, 964724959598400, 46628373047256000, 340112838697632000, 98292610383615648000
Offset: 1

Views

Author

Wolfdieter Lang, Jul 13 2007

Keywords

Comments

Numerators are given in A130549.
For the rationals r(n):= 2*sum(1/(j^2*binomial(2*j,j)),j=1..n), n>=1, the van der Poorten reference and a W. Lang link see A130551.

Links

Crossrefs

Cf. A112099, A112100, A112102, A112103, A130549.

Programs

  • Mathematica
    Table[2*Sum[1/(i^2*Binomial[2*i, i]), {i, 1, n}], {n, 1, 20}] // Denominator (* Vaclav Kotesovec, Mar 10 2016 *)
(2Accumulate[Table[1/(n^2 Binomial[2n,n]),{n,20}]])//Denominator (* Harvey P. Dale, Jan 27 2019 *)
  • PARI
    a(n) = denominator(2*sum(i=1, n, 1/(i^2*binomial(2*i, i)))); \\ Michel Marcus, Mar 10 2016

Formula

a(n) = denominator(r(n)), n>=1.
Denominator of 2*Sum_{i=1..n} 1/(i^2*C(2*i,i)). - Wolfdieter Lang, Oct 07 2008, corrected by Vaclav Kotesovec, Mar 10 2016

A130552 Denominators of partial sums for a series of (4/5)*Zeta(3).

Original entry on oeis.org

1, 24, 1080, 60480, 1512000, 7128000, 31783752000, 254270016000, 38903312448000, 67196630592000, 89438715317952000, 9308101594176000, 347648286440879424000, 347648286440879424000, 50409001533927516480000, 735378375318472005120000, 3612913957939652961154560000, 18401938665227434437120000
Offset: 1

Views

Author

Wolfdieter Lang, Jul 13 2007

Keywords

Comments

For the rationals r(n) := 2*Sum_{j=1..n} ((-1)^(j-1))/((j^3)*binomial(2*j,j)), n >= 1, the van der Poorten reference and a W. Lang link see A130551.
Numerators are given in A130551.

Crossrefs

Cf. A130551 (numerators).

Programs

  • PARI
    a(n) = denominator(2*sum(j=1, n, (-1)^(j-1)/(j^3*binomial(2*j,j)))); \\ Michel Marcus, Nov 09 2019

Formula

a(n) = denominator(r(n)), n >= 1.

Extensions

More terms from Michel Marcus, Nov 09 2019

A130557 Numerators of partial sums of a series for 6*(5 - 4*Zeta(3)).

Original entry on oeis.org

1, 10, 409, 10297, 8031, 394019, 9462581, 766743461, 8435956183, 1020884056543, 13272613316059, 2243198436149971, 2243285892433171, 2243347792046947, 305101392961615867, 88175602457796281563, 186150555360181760633
Offset: 2

Views

Author

Wolfdieter Lang, Jul 13 2007

Keywords

Comments

Denominators are given in A130558.
The rational sequence r(n) = 24*Sum_{j=2..n} 1/(j^3*(j^2-1)), n >= 2, tends, in the limit n->infinity, to 6*(5-4*Zeta(3)) which is approximately 1.15063433.

Examples

			Rationals r(n), n >= 2: 1, 10/9, 409/360, 10297/9000, 8031/7000, 394019/343000, ....

References

  • Z. A. Melzak, Companion to concrete mathematics,( Vol.I), Wiley, New York, 1973, pp. 83-84.

Links

Crossrefs

Cf. A130551/A130552 with the limit (4/5)*Zeta(3).

Formula

a(n) = numerator(r(n)), n >= 2, with the rationals r(n) defined above.
Showing 1-3 of 3 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.