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

0,1

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) = 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

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

Prog

(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))", "))
(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
(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

Crossrefs

Denominators are in A084226, decimal expansion is in A002117.

Cf. A084223 (s=2).

Sequence in context: A308491 A349901 A355469 * A278550 A177652 A278795

Adjacent sequences: A084222 A084223 A084224 * A084226 A084227 A084228

Keyword

nonn,frac

Author

Ralf Stephan, May 19 2003

Status

approved