A084223 -id:A084223 - cp's OEIS frontend

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

Ralf Stephan, May 19 2003

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.

Denominators are in A084226, decimal expansion is in A002117.

Cf. A084223 (s=2).

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

[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

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

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))","))

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

A084224 Denominators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=2.

Original entry on oeis.org

24, 1728, 324000, 19559232000, 208039104000, 181050031008000, 1889392861091736000, 32719838723847475200, 126909921829154720256000, 25243779460958994560841216000

Offset: 1

Ralf Stephan, May 19 2003

G. C. Greubel, Table of n, a(n) for n = 1..328
D. Zeilberger, Faster and Faster convergent series for zeta(3), arXiv:math/9804126 [math.CO], 1998.

Numerators are in A084223, decimal expansion is in A002117.

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))),DenominatorRat); # Muniru A Asiru, Oct 09 2018

[Denominator((&+[(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

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(denom(a(n)),n=1..10); # Muniru A Asiru, Oct 09 2018

Table[Denominator[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 *)

for(n=1,15,print1(denominator(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))), ","))

a(n) = denominator( 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

cp's OEIS Frontend

A084225 Numerators of successive approximations to zeta(3) = Sum_{k>0} 1/k^3, using Zeilberger's formula with s=3.

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