cp's OEIS Frontend

A084223 Numerators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=2.

%I A084223 #21 Jan 11 2025 20:34:36
%S A084223 29,2077,389467,23511309071,250074841297,217632439585619,
%T A084223 2271157731457180823,39331108008268763851,152552947614179997630583,
%U A084223 30344459362884140864563052777,40899053923887320978111449369,248809394545968517041811755878299,1150743449775104391337057432898154107
%N A084223 Numerators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=2.
%H A084223 G. C. Greubel, <a href="/A084223/b084223.txt">Table of n, a(n) for n = 1..329</a>
%H A084223 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 A084223 a(n) = numerator( 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 A084223 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(numer(a(n)),n=1..10); # _Muniru A Asiru_, Oct 09 2018
%t A084223 Table[Numerator[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 A084223 (PARI) for(n=1,15,print1(numerator(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 A084223 (Magma) [Numerator((&+[(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 A084223 (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))),NumeratorRat); # _Muniru A Asiru_, Oct 09 2018
%Y A084223 Denominators are in A084224, decimal expansion is in A002117.
%Y A084223 Cf. A084225 (s=3).
%K A084223 nonn,frac
%O A084223 1,1
%A A084223 _Ralf Stephan_, May 19 2003