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

29, 2077, 389467, 23511309071, 250074841297, 217632439585619, 2271157731457180823, 39331108008268763851, 152552947614179997630583, 30344459362884140864563052777, 40899053923887320978111449369, 248809394545968517041811755878299, 1150743449775104391337057432898154107

Offset: 1

Author

Ralf Stephan, May 19 2003

Keywords

• nonn
• frac

Links

• G. C. Greubel, Table of n, a(n) for n = 1..329
• D. Zeilberger, Faster and Faster convergent series for zeta(3), arXiv:math/9804126 [math.CO], 1998.

Crossrefs

Denominators are in A084224, decimal expansion is in A002117.

Cf. A084225 (s=3).

Programs

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

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

Maple

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

Mathematica

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 *)

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)), ","))

Formula

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