H := proc(n) local b;
b := s -> (-1)^(n+1)*(Zeta(n, 1-s, n+1) - s*Zeta(n+1, 1-s, n+1)/(n+1)); n!*limit(b(x), x=0) end:
seq(lprint([n], evalf(sort(simplify(H(n))))), n=0..20);

[0], -.5772156649
[1], 0.7281584548e-1
[2], .4998337404
[3], 3.638688872
[4], 36.52818814
[5], 486.2912247
[6], 8213.020078
[7], 169902.2977
[8], 4190128.695
[9], 120615178.1
[10], 3984800570.
[11], 0.1490393961e12
[12], 0.6239923527e13
[13], 0.2896860668e15
[14], 0.1479284230e17
[15], 0.8251797358e18
[16], 0.4998182098e20
[17], 0.3270091062e22
[18], 0.2300262764e24
[19], -0.6126113589e14
[20], -0.1134568187e16

[19] and [20] looks suspicious.
Remove the evalf to see the whole disaster.
Has this been fixed in the meanwhile?

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