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This is easy to do with PARI/GP. Here is my code

p(n) = binomial(n+2,2);
Y(k,b) = sumnum(l=0, (2*l+3)/(p(k)+p(l))^b,sumtable);
Z(a,b) = sumnum(k=0, (2*k+3)/p(k)^a*Y(k,b));
default(realprecision,57);
sumtable = sumnuminit();
print(2*Z(2,2)+4*Z(1,3))
/* 16.0000000000000000000000000000000000000000000000000000000 */

It takes 1.361 CPU seconds for 57 decimal places. The documentation has some information about the methods being used and there are several summation functions other than sumnum. For example, I originally used sumpos but sumnum is faster. Thanks to Henri Cohen for his comment to use sumnuminit to speed up the calculation of Y(k,b). Thanks to Jorge Zuniga for his comment that replacing sumnum with summonien is much faster. And especially thanks to the developers of PARI/GP who provided these fast numerical summation functions.

More details on Monien summation are in the arXiv paper Gaussian Summation: An Exponentially Converging Summation Scheme by Hartmut Monien. (published in Mathematics of Computation).

This is easy to do with PARI/GP. Here is my code

p(n) = binomial(n+2,2);
Y(k,b) = sumnum(l=0, (2*l+3)/(p(k)+p(l))^b,sumtable);
Z(a,b) = sumnum(k=0, (2*k+3)/p(k)^a*Y(k,b));
default(realprecision,57);
sumtable = sumnuminit();
print(2*Z(2,2)+4*Z(1,3))
/* 16.0000000000000000000000000000000000000000000000000000000 */

It takes 1.361 CPU seconds for 57 decimal places. The documentation has some information about the methods being used and there are several summation functions other than sumnum. For example, I originally used sumpos but sumnum is faster. Thanks to Henri Cohen for his comment to use sumnuminit to speed up the calculation of Y(k,b). Thanks to Jorge Zuniga for his comment that replacing sumnum with summonien is much faster. And especially thanks to the developers of PARI/GP who provided these fast numerical summation functions.

Some details are in the arXiv paper Gaussian Summation: An Exponentially Converging Summation Scheme by Hartmut Monien. (published in 2010 in Mathematics of Computation).

This is easy to do with PARI/GP. Here is my code

p(n) = binomial(n+2,2);
Y(k,b) = sumnum(l=0, (2*l+3)/(p(k)+p(l))^b,sumtable);
Z(a,b) = sumnum(k=0, (2*k+3)/p(k)^a*Y(k,b));
default(realprecision,57);
sumtable = sumnuminit();
print(2*Z(2,2)+4*Z(1,3))
/* 16.0000000000000000000000000000000000000000000000000000000 */

It takes 1.361 CPU seconds for 57 decimal places. The documentation has some information about the methods being used and there are several summation functions other than sumnum. For example, I originally used sumpos but sumnum is faster. Thanks to Henri Cohen for his comment to use sumnuminit to speed up the calculation of Y(k,b). Thanks to Jorge Zuniga for his comment that replacing sumnum with summonien is much faster. And especially thanks to the developers of PARI/GP who provided these fast numerical summation functions.

More details on Monien summation are in the paper Gaussian Summation: An Exponentially Converging Summation Scheme by Hartmut Monien.

This is easy to do with PARI/GP. Here is my code

p(n) = binomial(n+2,2);
Y(k,b) = sumnum(l=0, (2*l+3)/(p(k)+p(l))^b,sumtable);
Z(a,b) = sumnum(k=0, (2*k+3)/p(k)^a*Y(k,b));
default(realprecision,57);
sumtable = sumnuminit();
print(2*Z(2,2)+4*Z(1,3))
/* 16.0000000000000000000000000000000000000000000000000000000 */

It takes 1.361 CPU seconds for 57 decimal places. The documentation has some information about the methods being used and there are several summation functions other than sumnum. For example, I originally used sumpos but sumnum is faster. Thanks to Henri Cohen for his comment to use sumnuminit to speed up the calculation of Y(k,b). Thanks to Jorge Zuniga for his comment that replacing sumnum with summonien is much faster. And especially thanks to the developers of PARI/GP who provided these fast numerical summation functions.

More details on Monien summation are in the arXiv paper Gaussian Summation: An Exponentially Converging Summation Scheme by Hartmut Monien. (published in Mathematics of Computation).

This is easy to do with PARI/GP. Here is my code

p(n) = binomial(n+2,2);
Y(k,b) = sumnum(l=0, (2*l+3)/(p(k)+p(l))^b,sumtable);
Z(a,b) = sumnum(k=0, (2*k+3)/p(k)^a*Y(k,b));
default(realprecision,57);
sumtable = sumnuminit();
print(2*Z(2,2)+4*Z(1,3))
/* 16.0000000000000000000000000000000000000000000000000000000 */

It takes 1.361 CPU seconds for 57 decimal places. The documentation has some information about the methods being used and there are several summation functions other than sumnum. For example, I originally used sumpos but sumnum is faster. Thanks to Henri Cohen for his comment to use sumnuminit to speed up the calculation of Y(k,b). Thanks to Jorge Zuniga for his comment that replacing sumnum with summonien is much faster. And especially thanks to the developers of PARI/GP who provided these fast numerical summation functions.

This is easy to do with PARI/GP. Here is my code

p(n) = binomial(n+2,2);
Y(k,b) = sumnum(l=0, (2*l+3)/(p(k)+p(l))^b,sumtable);
Z(a,b) = sumnum(k=0, (2*k+3)/p(k)^a*Y(k,b));
default(realprecision,57);
sumtable = sumnuminit();
print(2*Z(2,2)+4*Z(1,3))
/* 16.0000000000000000000000000000000000000000000000000000000 */

It takes 1.361 CPU seconds for 57 decimal places. The documentation has some information about the methods being used and there are several summation functions other than sumnum. For example, I originally used sumpos but sumnum is faster. Thanks to Henri Cohen for his comment to use sumnuminit to speed up the calculation of Y(k,b). Thanks to Jorge Zuniga for his comment that replacing sumnum with summonien is much faster. And especially thanks to the developers of PARI/GP who provided these fast numerical summation functions.

More details on Monien summation are in the paper Gaussian Summation: An Exponentially Converging Summation Scheme by Hartmut Monien.