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

Offset

1,1

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.

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

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 *)

Prog

(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)), ", "))
(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
(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

Crossrefs

Denominators are in A084224, decimal expansion is in A002117.

Cf. A084225 (s=3).

Sequence in context: A042624 A190826 A045688 * A138755 A265445 A370339

Adjacent sequences: A084220 A084221 A084222 * A084224 A084225 A084226

Keyword

nonn,frac

Author

Ralf Stephan, May 19 2003

Status

approved