Lars Hesselholt.sagews

for n in [2..20]:
    print zeta(-n), -bernoulli(n+1)/(n+1)

0 0 1/120 1/120 0 0 -1/252 -1/252 0 0 1/240 1/240 0 0 -1/132 -1/132 0 0 691/32760 691/32760 0 0 -1/12 -1/12 0 0 3617/8160 3617/8160 0 0 -43867/14364 -43867/14364 0 0 174611/6600 174611/6600 0 0

for n in [2..20]:
    print n, bernoulli(n)

2 1/6 3 0 4 -1/30 5 0 6 1/42 7 0 8 -1/30 9 0 10 5/66 11 0 12 -691/2730 13 0 14 7/6 15 0 16 -3617/510 17 0 18 43867/798 19 0 20 -174611/330

taylor(x/(e^x-1), x, 0, 10)

1/47900160*x^10 - 1/1209600*x^8 + 1/30240*x^6 - 1/720*x^4 + 1/12*x^2 - 1/2*x + 1