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

0,1

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.

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

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 *)

Prog

(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))", "))
(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
(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

Crossrefs

Numerators are in A084225, decimal expansion is in A002117.

Sequence in context: A364304 A030254 A085482 * A071800 A093254 A347061

Adjacent sequences: A084223 A084224 A084225 * A084227 A084228 A084229

Keyword

nonn,frac

Author

Ralf Stephan, May 19 2003

Status

approved