A084225 Numerators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=3.
65, 25243, 211601801729, 41606201661907, 26719502723174333, 21470414849401610158757, 1743934446142768167359788693, 34556860353606738134995908106747
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..229
- D. Zeilberger, Faster and Faster convergent series for zeta(3), arXiv:math/9804126 [math.CO], 1998.
Programs
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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
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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
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Maple
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
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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))","))
Formula
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