A084224 - OEIS

%I #17 Sep 08 2022 08:45:11

%S 24,1728,324000,19559232000,208039104000,181050031008000,

%T 1889392861091736000,32719838723847475200,126909921829154720256000,

%U 25243779460958994560841216000

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

%H G. C. Greubel, <a href="/A084224/b084224.txt">Table of n, a(n) for n = 1..328</a>

%H D. Zeilberger, <a href="https://arxiv.org/abs/math/9804126">Faster and Faster convergent series for zeta(3)</a>, arXiv:math/9804126 [math.CO], 1998.

%F 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

%p 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

%t 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 *)

%o (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))), ","))

%o (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

%o (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

%Y Numerators are in A084223, decimal expansion is in A002117.

%K nonn,frac

%O 1,1

%A _Ralf Stephan_, May 19 2003