cp's OEIS Frontend

A084225 Numerators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=3.

%I A084225 #17 Sep 08 2022 08:45:11
%S A084225 65,25243,211601801729,41606201661907,26719502723174333,
%T A084225 21470414849401610158757,1743934446142768167359788693,
%U A084225 34556860353606738134995908106747
%N A084225 Numerators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=3.
%H A084225 G. C. Greubel, <a href="/A084225/b084225.txt">Table of n, a(n) for n = 0..229</a>
%H A084225 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 A084225 a(n) = numerator( 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 A084225 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(numer(a(n)),n=0..10); # _Muniru A Asiru_, Oct 09 2018
%o A084225 (PARI) for(n=0,10,print1(numerator(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 A084225 (Magma) [Numerator((&+[(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 A084225 (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))),NumeratorRat); # _Muniru A Asiru_, Oct 09 2018
%Y A084225 Denominators are in A084226, decimal expansion is in A002117.
%Y A084225 Cf. A084223 (s=2).
%K A084225 nonn,frac
%O A084225 0,1
%A A084225 _Ralf Stephan_, May 19 2003