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%I #13 Sep 08 2022 08:45:11
%S 54,21000,176033088000,34612505928000,22228151306961600,
%T 17861396405584738406400,1450791923043620377059840000,
%U 28748106901407399430780215360000
%N Denominators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=3.
%H G. C. Greubel, <a href="/A084226/b084226.txt">Table of n, a(n) for n = 0..229</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=0..n} ( (1/72)*(-1)^k*(5265*k^4 +13878*k^3 +13761*k^2+6120*k+1040)/(binomial(3*k,k)*binomial(4*k,k)*(4*k+1)*(4*k+3)*(k+1)*(3*k+1)^2*(3*k+2)^2) ) ). - _G. C. Greubel_, Oct 08 2018
%p a:=n->add((1/72)*(-1)^k*(5265*k^4+13878*k^3+13761*k^2+6120*k+1040)/(binomial(3*k,k)*binomial(4*k,k)*(4*k+1)*(4*k+3)*(k+1)*(3*k+1)^2*(3*k+2)^2),k=0..n): seq(denom(a(n)),n=0..10); # _Muniru A Asiru_, Oct 09 2018
%t Table[Denominator[Sum[(1/72)*(-1)^k*(5265*k^4 + 13878*k^3 + 13761*k^2 + 6120*k + 1040)/(Binomial[3*k, k]*Binomial[4*k, k]*(4*k + 1)*(4*k + 3)*(k + 1)*(3*k + 1)^2*(3*k + 2)^2), {k, 0, n}]], {n, 0, 30}] (* _G. C. Greubel_, Oct 08 2018 *)
%o (PARI) for(n=0,10,print1(denominator(sum(k=0,n,1/72*(-1)^k*(5265*k^4 +13878*k^3+13761*k^2+6120*k+1040)/binomial(3*k,k)/binomial(4*k,k)/(4*k+1)/(4*k+3)/(k+1)/(3*k+1)^2/(3*k+2)^2))","))
%o (Magma) [Denominator((&+[(1/72)*(-1)^k*(5265*k^4 +13878*k^3 +13761*k^2 +6120*k+1040)/(Binomial(3*k,k)*Binomial(4*k,k)*(4*k+1)*(4*k+3)*(k+1)*(3*k+1)^2*(3*k+2)^2): k in [0..n]])): n in [0..30]]; // _G. C. Greubel_, Oct 08 2018
%o (GAP) List(List([0..10],n->Sum([0..n],k->(1/72)*(-1)^k*(5265*k^4+13878*k^3+13761*k^2+6120*k+1040)/(Binomial(3*k,k)*Binomial(4*k,k)*(4*k+1)*(4*k+3)*(k+1)*(3*k+1)^2*(3*k+2)^2))),DenominatorRat); # _Muniru A Asiru_, Oct 09 2018
%Y Numerators are in A084225, decimal expansion is in A002117.
%K nonn,frac
%O 0,1
%A _Ralf Stephan_, May 19 2003
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