A084226 Denominators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=3.

54, 21000, 176033088000, 34612505928000, 22228151306961600, 17861396405584738406400, 1450791923043620377059840000, 28748106901407399430780215360000

Offset: 0

Author

Ralf Stephan, May 19 2003

Keywords

• nonn
• frac

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.

Crossrefs

Numerators are in A084225, decimal expansion is in A002117.

Programs

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

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

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(denom(a(n)),n=0..10); # Muniru A Asiru, Oct 09 2018

Mathematica

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 *)

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))","))

Formula

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