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308 bytes, as of the version from 2026-11-01 09:03 (current). Recorded here, not run.
import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
from math import comb
def c(k, digits=100):
F = RealBallField(numberdb.bits(digits, losing=k+80))
return sum(((-1)**j * comb(k,j) / F(2*j+2).zeta()
for j in range(k+1)), F(0))
[c(k) for k in range(257)]