A084224 Denominators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=2.

24, 1728, 324000, 19559232000, 208039104000, 181050031008000, 1889392861091736000, 32719838723847475200, 126909921829154720256000, 25243779460958994560841216000

Offset: 1

Author

Ralf Stephan, May 19 2003

Keywords

• nonn
• frac

Links

• G. C. Greubel, Table of n, a(n) for n = 1..328
• D. Zeilberger, Faster and Faster convergent series for zeta(3), arXiv:math/9804126 [math.CO], 1998.

Crossrefs

Numerators are in A084223, decimal expansion is in A002117.

Programs

GAP

List(List([1..10],n->Sum([1..n],k->(1/4)*(-1)^(k-1)*(56*k^2-32*k+5)/((2*k-1)^2*Binomial(3*k,k)*Binomial(2*k,k)*k^3))),DenominatorRat); # Muniru A Asiru, Oct 09 2018

Magma

[Denominator((&+[(1/4)*(-1)^(k-1)*(56*k^2-32*k+5)/((2*k-1)^2*Binomial(3*k,k)*Binomial(2*k,k)*k^3): k in [1..n]])): n in [1..30]]; // G. C. Greubel, Oct 08 2018

Maple

a:=n->add((1/4)*(-1)^(k-1)*(56*k^2-32*k+5)/((2*k-1)^2*binomial(3*k,k)*binomial(2*k,k)*k^3),k=1..n): seq(denom(a(n)),n=1..10); # Muniru A Asiru, Oct 09 2018

Mathematica

Table[Denominator[Sum[(1/4)*(-1)^(k-1)*(56*k^2-32*k+5)/((2*k-1)^2*Binomial[3*k,k]* Binomial[2*k,k]*k^3), {k,1,n}]], {n,1,30}] (* G. C. Greubel, Oct 08 2018 *)

PARI

for(n=1,15,print1(denominator(sum(k=1,n,(1/4)*(-1)^(k-1)*(56*k^2 -32*k +5)/((2*k-1)^2*binomial(3*k,k) *binomial(2*k,k)*k^3))), ","))

Formula

a(n) = denominator( Sum_{k=1..n} (1/4)*(-1)^(k-1)*(56*k^2-32*k+5)/((2*k-1)^2 * binomial(3*k,k) * binomial(2*k,k) * k^3) ). - G. C. Greubel, Oct 08 2018